Forecasting correlated time series with exponential smoothing models

This paper presents the Bayesian analysis of a general multivariate exponential smoothing model that allows us to forecast time series jointly, subject to correlated random disturbances. The general multivariate model, which can be formulated as a seemingly unrelated regression model, includes the previously studied homogeneous multivariate Holt-Winters’ model as a special case when all of the univariate series share a common structure. MCMC simulation techniques are required in order to approach the non-analytically tractable posterior distribution of the model parameters. The predictive distribution is then estimated using Monte Carlo integration. A Bayesian model selection criterion is introduced into the forecasting scheme for selecting the most adequate multivariate model for describing the behaviour of the time series under study. The forecasting performance of this procedure is tested using some real examples.     

Stochastic model specification search for Gaussian and partial non-Gaussian state space models

Model specification for state space models is a difficult task as one has to decide which components to include in the model and to specify whether these components are fixed or time-varying. To this aim a new model space MCMC method is developed in this paper. It is based on extending the Bayesian variable selection approach which is usually applied to variable selection in regression models to state space models. For non-Gaussian state space models stochastic model search MCMC makes use of auxiliary mixture sampling. We focus on structural time series models including seasonal components, trend or intervention. The method is applied to various well-known time series.     

贝叶斯视角下时变参数VAR建模——兼论“斜率之谜”

中国渐进式的改革实践要求中国宏观时间序列的建模能够允许参数平滑变化,而传统的VAR模型对此无能为力。本文详细阐述了在贝叶斯估计框架下,如何利用MCMC算法,建立时变参数VAR模型的过程,并利用该模型对徐高(2008)的数据重新进行了拟合,发现其文中提出的斜率之谜现象不复存在,因此时变参数VAR模型在拟合中国宏观时间序列方面更为精准。     

Stick-breaking autoregressive processes

This paper considers the problem of defining a time-dependent nonparametric prior for use in Bayesian nonparametric modelling of time series. A recursive construction allows the definition of priors whose marginals have a general stick-breaking form. The processes with Poisson-Dirichlet and Dirichlet process marginals are investigated in some detail. We develop a general conditional Markov Chain Monte Carlo (MCMC) method for inference in the wide subclass of these models where the parameters of the marginal stick-breaking process are nondecreasing sequences. We derive a generalised Pólya urn scheme type representation of the Dirichlet process construction, which allows us to develop a marginal MCMC method for this case. We apply the proposed methods to financial data to develop a semi-parametric stochastic volatility model with a time-varying nonparametric returns distribution. Finally, we present two examples concerning the analysis of regional GDP and its growth.     

Bayesian and Frequentist Inference for Ecological Inference: The R×C Case

In this paper we propose Bayesian and frequentist approaches to ecological inference, based on R × C contingency tables, including a covariate. The proposed Bayesian model extends the binomial-beta hierarchical model developed by K ing , R osen and T anner (1999) from the 2×2 case to the R × C case. As in the 2×2 case, the inferential procedure employs Markov chain Monte Carlo (MCMC) methods. As such, the resulting MCMC analysis is rich but computationally intensive. The frequentist approach, based on first moments rather than on the entire likelihood, provides quick inference via nonlinear least-squares, while retaining good frequentist properties. The two approaches are illustrated with simulated data, as well as with real data on voting patterns in Weimar Germany. In the final section of the paper we provide an overview of a range of alternative inferential approaches which trade-off computational intensity for statistical efficiency.     

The Beveridge–Nelson decomposition in retrospect and prospect

Beveridge and Nelson [Beveridge, Stephen, Nelson, Charles R., 1981. A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the ‘business cycle’. Journal of Monetary Economics 7, 151–174] proposed that the long-run forecast is a measure of trend for time series such as GDP that do not follow a deterministic path in the long run. They showed that if the series is stationary in first differences, then the estimated trend is a random walk with drift that accounts for growth, and the cycle is stationary. In contrast to linear de-trending, the smoother of Hodrick and Prescott (1981) and Hodrick and Prescott [Hodrick, Robert, Prescott, Edward C., 1997. Post-war US business cycles: An empirical investigation. Journal of Money Credit and Banking 29 (1), 1–16] and the unobserved components model of Harvey, [Harvey, A.C., 1985. Trends and cycles in macroeconomic time series. Journal of Business and Economic Statistics 3, 216–227]. Watson [Watson, Mark W., 1986. Univariate detrending methods with stochastic trends Journal of Monetary Economics 18, 49–75] and Clark [Clark, Peter K., 1987. The cyclical component of US economic activity. The Quarterly Journal of Economics 102 (4), 797–814], the BN decomposition attributes most variation in GDP to trend shocks while the cycles are short and brief. Since each is an estimate of the transitory part of GDP that will die out, it seems natural to compare cycle measures by their ability to forecast future growth. The results presented here suggest that cycle measures contain little if any information beyond the short-term momentum captured by BN.     

Analysis of high dimensional multivariate stochastic volatility models

This paper is concerned with the Bayesian estimation and comparison of flexible, high dimensional multivariate time series models with time varying correlations. The model proposed and considered here combines features of the classical factor model with that of the heavy tailed univariate stochastic volatility model. A unified analysis of the model, and its special cases, is developed that encompasses estimation, filtering and model choice. The centerpieces of the estimation algorithm (which relies on MCMC methods) are: (1) a reduced blocking scheme for sampling the free elements of the loading matrix and the factors and (2) a special method for sampling the parameters of the univariate SV process. The resulting algorithm is scalable in terms of series and factors and simulation-efficient. Methods for estimating the log-likelihood function and the filtered values of the time-varying volatilities and correlations are also provided. The performance and effectiveness of the inferential methods are extensively tested using simulated data where models up to 50 dimensions and 688 parameters are fit and studied. The performance of our model, in relation to various multivariate GARCH models, is also evaluated using a real data set of weekly returns on a set of 10 international stock indices. We consider the performance along two dimensions: the ability to correctly estimate the conditional covariance matrix of future returns and the unconditional and conditional coverage of the 5% and 1% value-at-risk (VaR) measures of four pre-defined portfolios.     

对经济增长的时间序列分析

时间序列分析在经济运用中作用十分明显。利用1980～2003年国内生产总值的相关资料,运用时间序列分析,应用SAS软件对经济增长时间序列进行模型识别、拟合、估计和预测,预测结果较为满意。而改革开放以来,投资在经济增长中的作用越来越明显,在对经济增长序列进行时间序列分析的同时,也结合回归分析建立经济增长和投资的回归-时间序列组合模型来进行分析。     

NONLINEAR TIME SERIES MODELS IN ECONOMICS

Abstract. In recent years there has been great interest in developing nonlinear extensions to the basic Autoregressive Integrated Moving Average model popularised by Box and Jenkins. Many of these have been in response to observed nonlinear behaviour in scientific areas such as electronic engineering, geology and oceanography and, as a consequence, have found little application in economics. Economic time series have features peculiar to themselves, and thus often require models to be developed in response to their own special nonlinear character. This paper therefore surveys those nonlinear time series models that have been developed in other disciplines and which have found to be useful for analysing economic time series, such as power transformations, fractional integration and deterministic chaos, and those that have been developed directly in response to nonlinear economic behaviour: for example, logistic transformations, asymmetric models, Markov models for business cycles and time deformation models. Also discussed are various tests for the presence of nonlinearity in time series and the evidence concerning the prevalence of such nonlinearity in economic time series is surveyed.     

COMMON SEASONAL FEATURES: GLOBAL UNEMPLOYMENT

Seasonal patterns in economic time series are generally examined from a univariate point of view. Using extensions of the unit root literature, important classes of seasonal processes are deterministic, stationary stochastic or mean reverting, and unit root stochastic. Time series tests have been developed for each of these. This paper examines seasonality in a multivariate context. Systems of economic variables can have trends, cycles and unit roots as well as the various types of seasonality. Restrictions such as cointegration and common cycles are here applied also to examine multivariate seasonal behaviour of economic variables. If each of a collection of series has a certain type of seasonality but a linear combination of these series can be found without seasonality, then the seasonal is said to be ‘common’. New tests are developed to determine if seasonal characteristics are common to a set of time series. These tests can be employed in the presence of various other time series structures. The analysis is applied to OECD data on unemployment for the period 1975.1 to 1993.4, and it is found that four diverse countries (Australia, Canada, Japan and USA) not only have common trends in their unemployment, but also have common deterministic seasonal features and a common cycle/stochastic seasonal feature. Such a collection of characteristics were not found in other groups of OECD countries.     

A CARBON FOOTPRINT TIME SERIES OF THE UK – RESULTS FROM A MULTI-REGION INPUT–OUTPUT MODEL

The framework and results of an international multi-region input–output (MRIO) model for the UK are presented. A time series of balanced input–output tables for the UK was constructed for the period 1992 to 2004 by using a matrix balancing procedure that is able to handle conflicting external data and inconsistent constraints. Detailed sectoral and country-specific trade data for the UK were compiled and reconciled with the UK input–output data, and economic and environmental accounts for three world regions were integrated in a UK-specific MRIO model. This was subsequently used to calculate a time series of national carbon footprints for the UK from 1992 to 2004. Greenhouse gas emissions embedded in UK trade are distinguished by destination of imports to intermediate and final demand. Most greenhouse gases show a significant increase over time in consumer emissions and a widening gap between producer and consumer emissions. Net CO₂ emissions embedded in UK imports increased from 4.3% of producer emissions in 1992 to a maximum of 20% in 2002. The total estimated UK carbon footprint in 2004 was 730 Mt for CO₂ and 934 Mt CO₂ equivalents for all greenhouse gases.     

基于Sieve Bootstrap方法的季节时间序列模型检验问题研究

研究目标：完善季节时间序列模型建模理论,解决建模过程烦琐、各类检验方法的结论差异大以及模型误设定问题。研究方法：基于对各季节时间序列模型的数理分析及比较,提出合理的模型检验程序;再运用Sieve Bootstrap方法,给出季节性单位根检验及确定性季节过程检验的统计量的临界值,并比较基于Sieve Bootstrap的检验方法与HEGY检验、BT检验的异同。研究发现：本文提出的检验程序能有效识别模型,检验统计量有限样本性质优良;实证分析表明,本文提出的检验程序及方法能更有效地识别中国宏观经济数据中的季节性。研究创新：将Sieve Bootstrap方法应用于季节时间序列的平稳性检验及趋势性检验中。研究价值：提出季节时间序列模型检验程序及检验方法,促进其在季节性经济数据中的应用。     

Forecasting economic time series with unconditional time-varying variance

The classical forecasting theory of stationary time series exploits the second-order structure (variance, autocovariance, and spectral density) of an observed process in order to construct some prediction intervals. However, some economic time series show a time-varying unconditional second-order structure. This article focuses on a simple and meaningful model allowing this nonstationary behaviour. We show that this model satisfactorily explains the nonstationary behaviour of several economic data sets, among which are the U.S. stock returns and exchange rates. The question of how to forecast these processes is addressed and evaluated on the data sets.     

A Bayesian analysis of market information linkages among NAFTA countries using a multivariate stochastic volatility model

NAFTA has arguably been the most important and elaborate free-trade agreement in history, providing a blueprint for potential new agreements. So far, the evidence is mixed as to whether NAFTA has been successful in terms of its economic impact. We fit a multivariate stochastic volatility model that directly measures financial information linkages across the three participating countries in a trivariate setting. The model detects significant changes in information linkages across the countries from the pre- to post-NAFTA period with a high degree of reliability. This has implications not only for measuring these linkages but also for hedging and portfolio diversification policies. An MCMC procedure is used to fit the model, and the accuracy and robustness of the method is confirmed by simulations.     

On the Stability of the Excess Sensitivity of Aggregate Consumption Growth in the USA

This paper investigates whether there is time variation in the excess sensitivity of aggregate consumption growth to anticipated aggregate disposable income growth using quarterly US data over the period 1953–2014. Our empirical framework contains the possibility of stickiness in aggregate consumption growth and takes into account measurement error and time aggregation. Our empirical specification is cast into a Bayesian state‐space model and estimated using Markov chain Monte Carlo (MCMC) methods. We use a Bayesian model selection approach to deal with the non‐regular test for the null hypothesis of no time variation in the excess sensitivity parameter. Anticipated disposable income growth is calculated by incorporating an instrumental variables estimation approach into our MCMC algorithm. Our results suggest that the excess sensitivity parameter in the USA is stable at around 0.23 over the entire sample period. Copyright © 2016 John Wiley & Sons, Ltd.     

A threshold stochastic volatility model with explanatory variables

In this paper, we introduce a threshold stochastic volatility model with explanatory variables. The Bayesian method is considered in estimating the parameters of the proposed model via the Markov chain Monte Carlo (MCMC) algorithm. Gibbs sampling and Metropolis–Hastings sampling methods are used for drawing the posterior samples of the parameters and the latent variables. In the simulation study, the accuracy of the MCMC algorithm, the sensitivity of the algorithm for model assumptions, and the robustness of the posterior distribution under different priors are considered. Simulation results indicate that our MCMC algorithm converges fast and that the posterior distribution is robust under different priors and model assumptions. A real data example was analyzed to explain the asymmetric behavior of stock markets.     

Bayesian estimation and model selection for spatial Durbin error model with finite distributed lags

In this paper we investigate a spatial Durbin error model with finite distributed lags and consider the Bayesian MCMC estimation of the model with a smoothness prior. We study also the corresponding Bayesian model selection procedure for the spatial Durbin error model, the spatial autoregressive model and the matrix exponential spatial specification model. We derive expressions of the marginal likelihood of the three models, which greatly simplify the model selection procedure. Simulation results suggest that the Bayesian estimates of high order spatial distributed lag coefficients are more precise than the maximum likelihood estimates. When the data is generated with a general declining pattern or a unimodal pattern for lag coefficients, the spatial Durbin error model can better capture the pattern than the SAR and the MESS models in most cases. We apply the procedure to study the effect of right to work (RTW) laws on manufacturing employment.     

NONLINEAR BUSINESS CYCLE MODELLING

Abstract. Literature which employs nonlinearities to explain economic fluctuations, commonly called business cycles, is surveyed. Relaxation of the linearity assumption significantly increases the range of possible dynamic solution paths and introduces the possibility that business cycles are endogenously determined. The dominant post-war modelling strategy has been the Frisch (1933) (and Slutsky, 1937) inspired one of developing essentially (log) linear economic models which produce damped cycles (or monotonic damping) to propagate the energy provided by repeated random (or autocorrelated) shocks. The cycle is exogenously driven, since it would die out in the absence of shocks. Deterministic (nonstochastic) nonlinear models can produce a wide range of endogenous fluctuations, including: stable limit cycles; growth cycles; and chaotic output, which have the appearance of random fluctuations. Further, the same model can produce qualitatively different outputs according to starting and parameter values. If the possibility of shocks to parameters is admitted, then behaviour can change abruptly following shocks. Evidence on the existence of nonlinearities and chaos in macroeconomic time series is assessed and alternative approaches to modelling dynamic economic development, related to the work of Keynes, Marx, Schumpeter and Shackle, are discussed. Their ideas have not proved readily amenable to mathematical modelling, but attempts to encapsulate some of them are reviewed.     

Growth, cycles and convergence in US regional time series

This article reports the results of fitting unobserved components (structural) time series models to data on real income per capita in eight regions of the United States. The aim is to establish stylised facts about cycles and convergence. It appears that while the cycles are highly correlated, the two richest regions have been diverging from the others in recent years. A new model is developed in order to characterise the converging behaviour of the six poorest regions. The model combines convergence components with a common trend and cycles. These convergence components are formulated as a second-order error correction mechanism which allows temporary divergence while imposing eventual convergence. After fitting the model, the implications for forecasting are examined. Finally, the use of unit root tests for testing convergence is critically assessed in the light of the stylised facts obtained from the fitted models.     

Semiparametric time series regression modeling with a diverging number of parameters

Variable selection and error structure determination of a partially linear model with time series errors are important issues. In this paper, we investigate the regression coefficient and autoregressive order shrinkage and selection via the smoothly clipped absolute deviation penalty for a partially linear model with a divergent number of covariates and finite order autoregressive time series errors. Both consistency and asymptotic normality of the proposed penalized estimators are derived. The oracle property of the resultant estimators is proved. Simulation studies are carried out to assess the finite‐sample performance of the proposed procedure. A real data analysis is made to illustrate the usefulness of the proposed procedure as well.