Real-Time Forecasting With a Mixed-Frequency VAR

This article develops a vector autoregression (VAR) for time series which are observed at mixed frequencies—quarterly and monthly. The model is cast in state-space form and estimated with Bayesian methods under a Minnesota-style prior. We show how to evaluate the marginal data density to implement a data-driven hyperparameter selection. Using a real-time dataset, we evaluate forecasts from the mixed-frequency VAR and compare them to standard quarterly frequency VAR and to forecasts from MIDAS regressions. We document the extent to which information that becomes available within the quarter improves the forecasts in real time. This article has online supplementary materials.     

ANALYSIS OF VECTOR AUTOREGRESSIONS IN THE PRESENCE OF SHIFTS IN MEAN

This paper considers the implications of mean shifts in a multivariate setting. It is shown that under the additive outlier type mean shift specification, the intercept in each equation of the vector autoregression (VAR) will be subject to multiple shifts when the break dates of the mean shifts to the univariate series do not coincide. Conversely, under the innovative outlier type mean shift specification, both the univariate and the multivariate time series are subject to multiple shifts when mean shifts to the innovation processes occur at different dates. We consider two procedures, the first removes the shifts series by series before forming the VAR, and the second removes intercept shifts in the VAR directly. The pros and cons of both methods are discussed.     

Cross-entropy method for estimation of posterior expectation in Bayesian VAR models

In this article, an importance sampling (IS) method for the posterior expectation of a non linear function in a Bayesian vector autoregressive (VAR) model is developed. Most Bayesian inference problems involve the evaluation of the expectation of a function of interest, usually a non linear function of the model parameters, under the posterior distribution. Non linear functions in Bayesian VAR setting are difficult to estimate and usually require numerical methods for their evaluation. A weighted IS estimator is used for the evaluation of the posterior expectation. With the cross-entropy (CE) approach, the IS density is chosen from a specified family of densities such that the CE distance or the Kullback–Leibler divergence between the optimal IS density and the importance density is minimal. The performance of the proposed algorithm is assessed in an iterated multistep forecasting of US macroeconomic time series.     

Likelihood Function and Canonical Correlation Analysis of the Peña–Box Model

The Peña–Box model is a type of dynamic factor model whose factors try to capture the time-effect movements of a multiple time series. The Peña–Box model can be expressed as a vector autoregressive (VAR) model with constraints. This article derives the maximum likelihood estimates and the likelihood ratio test of the VAR model for Gaussian processes. Then a test statistic constructed by canonical correlation coefficients is presented and adjusted for conditional heteroscedasticity. Simulations confirm the validity of adjustments for conditional heteroscedasticity, and show that the proposed statistics perform better than the statistics used in the existing literature.     

On the estimation problem of periodic autoregressive time series: symmetric and asymmetric innovations

Periodic autoregressive (PAR) models with symmetric innovations are widely used on time series analysis, whereas its asymmetric counterpart inference remains a challenge, because of a number of problems related to the existing computational methods. In this paper, we use an interesting relationship between periodic autoregressive and vector autoregressive (VAR) models to study maximum likelihood and Bayesian approaches to the inference of a PAR model with normal and skew-normal innovations, where different kinds of estimation methods for the unknown parameters are examined. Several technical difficulties which are usually complicated to handle are reported. Results are compared with the existing classical solutions and the practical implementations of the proposed algorithms are illustrated via comprehensive simulation studies. The methods developed in the study are applied and illustrate a real-time series. The Bayes factor is also used to compare the multivariate normal model versus the multivariate skew-normal model.     

Structural vector autoregressive analysis for cointegrated variables

Summary Vector autoregressive (VAR) models are capable of capturing the dynamic structure of many time series variables. Impulse response functions are typically used to investigate the relationships between the variables included in such models. In this context the relevant impulses or innovations or shocks to be traced out in an impulse response analysis have to be specified by imposing appropriate identifying restrictions. Taking into account the cointegration structure of the variables offers interesting possibilities for imposing identifying restrictions. Therefore VAR models which explicitly take into account the cointegration structure of the variables, so-called vector error correction models, are considered. Specification, estimation and validation of reduced form vector error correction models is briefly outlined and imposing structural short- and long-run restrictions within these models is discussed. I thank an anonymous reader for comments on an earlier draft of this paper that helped me to improve the exposition.     

Dynamic Forecasts of Qualitative Variables

This article presents a new Qual VAR model for incorporating information from qualitative and/or discrete variables in vector autoregressions. With a Qual VAR, it is possible to create dynamic forecasts of the qualitative variable using standard VAR projections. Previous forecasting methods for qualitative variables, in contrast, produce only static forecasts. I apply the Qual VAR to forecasting the 2001 business recession out of sample and to analyzing the Romer and Romer narrative measure of monetary policy contractions as an endogenous variable in a VAR. Out of sample, the model predicts the timing of the 2001 recession quite well relative to the recession probabilities put forth at the time by professional forecasters. Qual VARs—which include information about the qualitative variable—can also enhance the quality of density forecasts of the other variables in the system.     

财政支农、农村居民消费与农民增收的动态分析——以辽宁省为例

政府财政支出和居民消费的关系一直是经济界争论的焦点,文章基于1980-2007年辽宁省的时间序列数据,从宏观和动态的角度,运用单位根检验,Granger因果检验和向量自回归模型考察辽宁省财政支农,农村居民消费以及收入之间的关系,向量自回归模型的动态分析表明：农村居民收入和消费之间具有相互促进的正向联系,而财政支出的各方面对居民收入和消费的影响不尽相同,其中,基本建设支出对居民消费和收入都有负影响．即产生“挤出效应”,而生产性支出对居民收入产生正影响,而对居民消费却产生“挤出效应”,科技项目支出对居民收入和消费,先产生负影响,随着时间的推移,影响变为正;其他财政支出对居民收入产生正的影响,而对居民消费先产生“挤出效应”,随后又促进居民消费。     

Analyzing multiple vector autoregressions through matrix-variate normal distribution with two covariance matrices

Abstract

This article proposes a new approach to analyze multiple vector autoregressive (VAR) models that render us a newly constructed matrix autoregressive (MtAR) model based on a matrix-variate normal distribution with two covariance matrices. The MtAR is a generalization of VAR models where the two covariance matrices allow the extension of MtAR to a structural MtAR analysis. The proposed MtAR can also incorporate different lag orders across VAR systems that provide more flexibility to the model. The estimation results from a simulation study and an empirical study on macroeconomic application show favorable performance of our proposed models and method.     

时间序列的图模型及其在股市相关性中的应用

图模型方法是高维数据统计分析的重要工具,时间序列的图模型方法有链图、因果图和偏相关图,将基于VAR模型的时间序列链图和因果图应用于国际股票市场,研究主要股指的动态相关性,结果表明：美国股市对周边股市的影响较大。将偏相关图应用于亚洲股票市场,研究亚洲主要股指的交互作用,结果表明：中国内地是相对独立的市场,中国香港、台湾以及新加坡、日本股票市场之间存在显著的信息流动。     

Bayesian averaging of classical estimates in asymmetric vector autoregressive models

The estimated vector autoregressive (VAR) model is sensitive to model misspecifications, resulting to biased and inconsistent parameter estimates. This article extends the Bayesian averaging of classical estimates, a robustness procedure in cross-section data, to a vector time-series that is estimated using a large number of asymmetric VAR models. The proposed procedure was applied to simulated data from various forms of model misspecifications. The results of the simulation suggest that, under misspecification problems, particularly if an important variable and moving average (MA) terms were omitted, the proposed procedure gives robust results and better forecasts than the automatically selected equal lag-length VAR model.     

Stochastic Model Specification Search for Time-Varying Parameter VARs

This article develops a new econometric methodology for performing stochastic model specification search (SMSS) in the vast model space of time-varying parameter vector autoregressions (VARs) with stochastic volatility and correlated state transitions. This is motivated by the concern of overfitting and the typically imprecise inference in these highly parameterized models. For each VAR coefficient, this new method automatically decides whether it is constant or time-varying. Moreover, it can be used to shrink an otherwise unrestricted time-varying parameter VAR to a stationary VAR, thus providing an easy way to (probabilistically) impose stationarity in time-varying parameter models. We demonstrate the effectiveness of the approach with a topical application, where we investigate the dynamic effects of structural shocks in government spending on U.S. taxes and gross domestic product (GDP) during a period of very low interest rates.     

Influence diagnostics in a vector autoregressive model

In this paper, we use a likelihood approach and the local influence method introduced by Cook [Assessment of local influence (with discussion). J Roy Statist Soc Ser B. 1986;48:133–149] to study a vector autoregressive (VAR) model. We present the maximum likelihood estimators and the information matrix. We establish the normal curvature and slope diagnostics for the VAR model under several perturbation schemes and use the Monte Carlo method to obtain benchmark values for determining the influence of directional diagnostics and possible influential observations. An empirical study using the VAR model to fit real data of monthly returns of IBM and S&P500 index illustrates the effectiveness of our proposed diagnostics.     

COMPOSITIONAL TIME SERIES ANALYSIS OF MORTALITY PROPORTIONS

Compositional time series are multivariate time series which at each time point are proportions that sum to a constant. Accurate inference for such series which occur in several disciplines such as geology, economics and ecology is important in practice. Usual multivariate statistical procedures ignore the inherent constrained nature of these observations as parts of a whole and may lead to inaccurate estimation and prediction. In this article, a regression model with vector autoregressive moving average (VARMA) errors is fit to the compositional time series after an additive log ratio (ALR) transformation. Inference is carried out in a hierarchical Bayesian framework using Markov chain Monte Carlo techniques. The approach is illustrated on compositional time series of mortality events in Los Angeles in order to investigate dependence of different categories of mortality on air quality.     

A multivariate long-memory model with structural breaks

This paper introduces a multivariate long-memory model with structural breaks. In the proposed framework, the time series exhibits possibly fractional orders of integration which are allowed to be different in each subsample. The break date is endogenously determined using a procedure that minimizes the residual sum of squares (RSS). Monte Carlo experiments show that this method for detecting breaks performs well in large samples. As an illustration, we estimate a trivariate VAR including prices, employment and GDP in both the US and Mexico. For the subsample preceding the break, our findings are similar to those of earlier studies based on a standard VAR approach in both the countries, such that the variables exhibit integer degrees of integration. On the contrary, the series is found to be fractionally integrated after the break, with the fractional differencing parameters being higher than one in the case of Mexico.     

Model selection criteria for reduced rank multivariate time series: a simulation study

The main focus of our paper is to compare the performance of different model selection criteria used for multivariate reduced rank time series. We consider one of the most commonly used reduced rank model, that is, the reduced rank vector autoregression (RRVAR (p, r)) introduced by Velu et al. [Reduced rank models for multiple time series. Biometrika. 1986;7(31):105–118]. In our study, the most popular model selection criteria are included. The criteria are divided into two groups, that is, simultaneous selection and two-step selection criteria, accordingly. Methods from the former group select both an autoregressive order p and a rank r simultaneously, while in the case of two-step criteria, first an optimal order p is chosen (using model selection criteria intended for the unrestricted VAR model) and then an optimal rank r of coefficient matrices is selected (e.g. by means of sequential testing). Considered model selection criteria include well-known information criteria (such as Akaike information criterion, Schwarz criterion, Hannan–Quinn criterion, etc.) as well as widely used sequential tests (e.g. the Bartlett test) and the bootstrap method. An extensive simulation study is carried out in order to investigate the efficiency of all model selection criteria included in our study. The analysis takes into account 34 methods, including 6 simultaneous methods and 28 two-step approaches, accordingly. In order to carefully analyse how different factors affect performance of model selection criteria, we consider over 150 simulation settings. In particular, we investigate the influence of the following factors: time series dimension, different covariance structure, different level of correlation among components and different level of noise (variance). Moreover, we analyse the prediction accuracy concerned with the application of the RRVAR model and compare it with results obtained for the unrestricted vector autoregression. In this paper, we also present a real data application of model selection criteria for the RRVAR model using the Polish macroeconomic time series data observed in the period 1997–2007.     

Bayes Estimates of Markov Trends in Possibly Cointegrated Series

Stylized facts show that average growth rates of U.S. per capita consumption and income differ in recession and expansion periods. Because a linear combination of such series does not have to be a constant mean process, standard cointegration analysis between the variables to examine the permanent income hypothesis may not be valid. To model the changing growth rates in both series, we introduce a multivariate Markov trend model that accounts for different growth rates in consumption and income during expansions and recessions and across variables within both regimes. The deviations from the multivariate Markov trend are modeled by a vector autoregression (VAR) model. Bayes estimates of this model are obtained using Markov chain Monte Carlo methods. The empirical results suggest the existence of a cointegration relation between U.S. per capita disposable income and consumption, after correction for a multivariate Markov trend. This result is also obtained when per capita investment is added to the VAR.     

Statistical Analysis Of Mixture Vector Autoregressive Models

In this paper, we reconsider the mixture vector autoregressive model, which was proposed in the literature for modelling non‐linear time series. We complete and extend the stationarity conditions, derive a matrix formula in closed form for the autocovariance function of the process and prove a result on stable vector autoregressive moving‐average representations of mixture vector autoregressive models. For these results, we apply techniques related to a Markovian representation of vector autoregressive moving‐average processes. Furthermore, we analyse maximum likelihood estimation of model parameters by using the expectation–maximization algorithm and propose a new iterative algorithm for getting the maximum likelihood estimates. Finally, we study the model selection problem and testing procedures. Several examples, simulation experiments and an empirical application based on monthly financial returns illustrate the proposed procedures.     

Asymmetric Forecast Densities for U.S. Macroeconomic Variables from a Gaussian Copula Model of Cross-Sectional and Serial Dependence

Most existing reduced-form macroeconomic multivariate time series models employ elliptical disturbances, so that the forecast densities produced are symmetric. In this article, we use a copula model with asymmetric margins to produce forecast densities with the scope for severe departures from symmetry. Empirical and skew t distributions are employed for the margins, and a high-dimensional Gaussian copula is used to jointly capture cross-sectional and (multivariate) serial dependence. The copula parameter matrix is given by the correlation matrix of a latent stationary and Markov vector autoregression (VAR). We show that the likelihood can be evaluated efficiently using the unique partial correlations, and estimate the copula using Bayesian methods. We examine the forecasting performance of the model for four U.S. macroeconomic variables between 1975:Q1 and 2011:Q2 using quarterly real-time data. We find that the point and density forecasts from the copula model are competitive with those from a Bayesian VAR. During the recent recession the forecast densities exhibit substantial asymmetry, avoiding some of the pitfalls of the symmetric forecast densities from the Bayesian VAR. We show that the asymmetries in the predictive distributions of GDP growth and inflation are similar to those found in the probabilistic forecasts from the Survey of Professional Forecasters. Last, we find that unlike the linear VAR model, our fitted Gaussian copula models exhibit nonlinear dependencies between some macroeconomic variables. This article has online supplementary material.     

Linear and nonlinear causality tests in an LSTAR model: wavelet decomposition in a nonlinear environment

In this paper, we use simulated data to investigate the power of different causality tests in a two-dimensional vector autoregressive (VAR) model. The data are presented in a nonlinear environment that is modelled using a logistic smooth transition autoregressive function. We use both linear and nonlinear causality tests to investigate the unidirection causality relationship and compare the power of these tests. The linear test is the commonly used Granger causality F test. The nonlinear test is a non-parametric test based on Baek and Brock [A general test for non-linear Granger causality: Bivariate model. Tech. Rep., Iowa State University and University of Wisconsin, Madison, WI, 1992] and Hiemstra and Jones [Testing for linear and non-linear Granger causality in the stock price–volume relation, J. Finance 49(5) (1994), pp. 1639–1664]. When implementing the nonlinear test, we use separately the original data, the linear VAR filtered residuals, and the wavelet decomposed series based on wavelet multiresolution analysis. The VAR filtered residuals and the wavelet decomposition series are used to extract the nonlinear structure of the original data. The simulation results show that the non-parametric test based on the wavelet decomposition series (which is a model-free approach) has the highest power to explore the causality relationship in nonlinear models.