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.     

Long-Run Identification in a Fractionally Integrated System

We propose an extension of structural fractionally integrated vector autoregressive models that avoids certain undesirable effects on the impulse responses that occur if long-run identification restrictions are imposed. We derive the model’s Granger representation and investigate the effects of long-run restrictions. Simulations illustrate that enforcing integer integration orders can have severe consequences for impulse responses. In a system of U.S. real output and aggregate prices, the effects of structural shocks strongly depend on the specification of the integration orders. In the statistically preferred fractional model, shocks that are typically interpreted as demand disturbances have a very brief influence on GDP. Supplementary materials for this article are available online.     

Reducing confidence bands for simulated impulse responses

It is emphasized that the shocks in structural vector autoregressions are only identified up to sign and it is pointed out that this feature can result in very misleading confidence intervals for impulse responses if simulation methods such as Bayesian or bootstrap methods are used. The confidence intervals heavily depend on which variable is used for fixing the signs of the responses. In particular, when the shocks are identified via long-run restrictions the problem can be severe. It is pointed out that a suitable choice of variable for fixing the signs of the responses and, hence, of the shocks, can result in substantial reductions in the confidence bands for impulse responses.     

Aggregate consumption functions for India: A cointegration analysis under structural changes, 1919-86

This paper extends cointegration methodology to include the effect of possible structural changes on aggregate consumption behaviour in India during 1919-86. The only cointegrated relation is found to be a dynamic linear regression of lag order two, with 1944 as the year in which structural change began. The estimated short-run marginal propensity to consume (MPC) is greater than the long-run MPC. The estimates of the MPC are different from previous estimates for the Indian economy based on conventional econometrics. The initial year of structural change has been selected by extending the method of Perron and that of Zivot and Andrews.     

Adaptive Shrinkage in Bayesian Vector Autoregressive Models

Vector autoregressive (VAR) models are frequently used for forecasting and impulse response analysis. For both applications, shrinkage priors can help improving inference. In this article, we apply the Normal-Gamma shrinkage prior to the VAR with stochastic volatility case and derive its relevant conditional posterior distributions. This framework imposes a set of normally distributed priors on the autoregressive coefficients and the covariance parameters of the VAR along with Gamma priors on a set of local and global prior scaling parameters. In a second step, we modify this prior setup by introducing another layer of shrinkage with scaling parameters that push certain regions of the parameter space to zero. Two simulation exercises show that the proposed framework yields more precise estimates of model parameters and impulse response functions. In addition, a forecasting exercise applied to U.S. data shows that this prior performs well relative to other commonly used specifications in terms of point and density predictions. Finally, performing structural inference suggests that responses to monetary policy shocks appear to be reasonable.     

Testing for nonzero impulse responses in vector autoregressive processes

Impulse response functions are often used to investigate the relationships between the components of a VAR (vector autoregressive) process. A hypothesis of particular interest is that a variable does not react to impulses in another variable, i.e., the impulse responses are zero. Two types of tests for such hypotheses are considered. The first type is based on finite-order VAR assumptions and the second allows for possibly infinite-order processes. It is found that both types of tests have to be used cautiously because small sample and asymptotic properties may differ substantially.     

Neglecting structural breaks when estimating and valuing dynamic correlations for asset allocation

This paper assesses the econometric and economic value consequences of neglecting structural breaks in dynamic correlation models and in the context of asset allocation framework. It is shown that changes in the parameters of the conditional correlation process can lead to biased estimates of persistence. Monte Carlo simulations reveal that short-run persistence is downward biased while long-run persistence is severely upward biased, leading to spurious high persistence of shocks to conditional correlation. An application to stock returns supports these results and concludes that neglecting such structural shifts could lead to misleading decisions on portfolio diversification, hedging, and risk management.     

Cointegration and Long-Run Asset Allocation

We show that economic restrictions of cointegration between asset cash flows and aggregate consumption have important implications for return dynamics and optimal portfolio rules, particularly at long investment horizons. When cash flows and consumption share a common stochastic trend (i.e., are cointegrated), temporary deviations between their levels forecast long-horizon dividend growth rates and returns, and consequently, alter the term profile of risks and expected returns. We show that the optimal asset allocation based on the error-correction vector autoregression (EC-VAR) specification can be quite different relative to a traditional VAR that ignores the cointegrating relation. Unlike the EC-VAR, the commonly used VAR approach to model expected returns focuses on short-run forecasts and can considerably miss on long-horizon return dynamics, and hence, the optimal portfolio mix in the presence of cointegration. We develop and implement methods to account for parameter uncertainty in the EC-VAR setup and highlight the importance of the error-correction channel for optimal portfolio decisions at various investment horizons.     

Short- and long-run rolling causality techniques and optimal window-wise lag selection: an application to the export-led growth hypothesis

The literature devoted to the export-led growth (ELG) hypothesis, which is of utmost importance for policymaking in emerging countries, provides mixed evidence for the validity of the hypothesis. Recent contributions focus on the time-dependence of the relationship between export and output growth using rolling causality techniques based on vector autoregressive models. These models focus on a short-term view which captures single policy-induced developments. However, long-term structural changes cannot be covered by examinations related to the short-term. This paper hence examines the time-varying validity of the ELG hypothesis for India for the period 1960–2011 using rolling causality techniques for both the short-run and long-run horizon. For the first time, window-wise optimal lag-selection procedures are applied in connection with these techniques. We find that exports long-run caused output growth from 1997 until 2009 which can be seen as a consequence of political reforms of the 1990s that boosted economic growth by generating foreign direct investment opportunities and higher exports. For the short-run, export significantly caused output in the period 1998–2003 which followed a concentration of liberalization measures in 1997. Causality in the reversed direction, from output to exports, only seems to be relevant in the short-run.     

Confidence intervals for impulse responses under departures from normality

Monte Carlo evidence shows that in structural VAR models with fat-tailed or skewed innovations the coverage accuracy of impulse response confidence intervals may deterorate substantially compared to the same model with Gaussian innovations. Empirical evidance suggests that such departures from normality are quite plausible for economic time series. The simulation results suggest that applied researchers are best off using nonparametric bootstrap intervals for impulse responses, regardless of whether or not there is evidence of fat tails or skewness in the error distribution. Allowing for departures from normality is shown to considerably weaken the evidence of the delayed overshooting puzzle in Eichenbaum and Evans (1995).     

Do Flow Rates Respond Asymmetrically to Water Level? Evidence from the Edwards Aquifer

This research examines the time series relationship between the Comal Springs flow rate and the water level in the Edwards Aquifer (Well J-17). The empirical methodology utilizes threshold autoregression (TAR) and momentum-TAR models that allow for asymmetry in responses and adjustments to a disequilibrium in the long-run cointegrating relationship. Based on the results, an asymmetric error-correction model (AECM) is proposed to characterize the short-run and long-run dynamic relationship between spring flow and water level. The results have implications for the management of water resources, water demand, and ecosystems.     

Estimation of long-run inefficiency levels: a dynamic frontier approach

Cornwell, Schmidt, and Sickles (1990) and Kumbhakar (1990), among others, developed stochasticfrontier production models which allow firm specific inefficiency levels to change over time. These studies assumed arbitrary restrictions on the short-run dynamics of efficiency levels which have little theoretical justification. Further, the models are inappropriate for estimation of long-run efficiencies. We consider estimation of an alternative frontier model in which firmspecific technical inefficiency levels are autoregressive. This model is particularly useful to examine a potential dynamic link between technical innovations and production inefficiency levels. We apply our methodology to a panel of US airlines.     

Estimation of long-run inefficiency levels: a dynamic frontier approach

Cornwell, Schmidt, and Sickles (1990) and Kumbhakar (1990), among others, developed stochasticfrontier production models which allow firm specific inefficiency levels to change over time. These studies assumed arbitrary restrictions on the short-run dynamics of efficiency levels which have little theoretical justification. Further, the models are inappropriate for estimation of long-run efficiencies. We consider estimation of an alternative frontier model in which firmspecific technical inefficiency levels are autoregressive. This model is particularly useful to examine a potential dynamic link between technical innovations and production inefficiency levels. We apply our methodology to a panel of US airlines.     

Some Observations,a Suggestion,and Some Comments on the Conway-Roberts Article

This article provides new tools for the evaluation of dynamic stochastic general equilibrium (DSGE) models and applies them to a large-scale new Keynesian model. We approximate the DSGE model by a vector autoregression, and then systematically relax the implied cross-equation restrictions and document how the model fit changes. We also compare the DSGE model's impulse responses to structural shocks with those obtained after relaxing its restrictions. We find that the degree of misspecification in this large-scale DSGE model is no longer so large as to prevent its use in day-to-day policy analysis, yet is not small enough to be ignored.     

LONG-RUN STRUCTURAL MODELLING

The paper develops a general framework for identification, estimation, and hypothesis testing in cointegrated systems when the cointegrating coefficients are subject to (possibly) non-linear and cross-equation restrictions, obtained from economic theory or other relevant a priori information. It provides a proof of the consistency of the quasi maximum likelihood estimators (QMLE), establishes the relative rates of convergence of the QMLE of the short-run and the long-run parameters, and derives their asymptotic distributions; thus generalizing the results already available in the literature for the linear case. The paper also develops tests of the over-identifying (possibly) non-linear restrictions on the cointegrating vectors. The estimation and hypothesis testing procedures are applied to an Almost Ideal Demand System estimated on U.K. quarterly observations. Unlike many other studies of consumer demand this application does not treat relative prices and real per capita expenditures as exogenously given.     

Comment on “recent developments in bootstrapping time series”

The paper develops a general framework for identification, estimation, and hypothesis testing in cointegrated systems when the cointegrating coefficients are subject to (possibly) non-linear and cross-equation restrictions, obtained from economic theory or other relevant a priori information. It provides a proof of the consistency of the quasi maximum likelihood estimators (QMLE), establishes the relative rates of convergence of the QMLE of the short-run and the long-run parameters, and derives their asymptotic distributions; thus generalizing the results already available in the literature for the linear case. The paper also develops tests of the over-identifying (possibly) non-linear restrictions on the cointegrating vectors. The estimation and hypothesis testing procedures are applied to an Almost Ideal Demand System estimated on U.K. quarterly observations. Unlike many other studies of consumer demand this application does not treat relative prices and real per capita expenditures as exogenously given.     

Valid Bayesian Estimation of the Cointegrating Error Correction Model

Two methods of identifying cointegrating vectors are commonly used: linear restrictions and the nonlinear method of Johansen's maximum likelihood procedure. That the linear method can produce invalid estimates while the Johansen approach always produces valid estimates has been recognized in several recent articles. Because all Bayesian studies to date have used linear restrictions, this article presents a Bayesian method for obtaining estimates of cointegrating vectors that will always be valid. In addition, it also presents an approach for evaluating the validity of linear restrictions.     

中国省际经济差距来源分析

在利用索洛模型对中国省区1985—2009年经济增长进行分解的基础上,结合空间标准差指标并采用基于VAR模型的脉冲响应函数分析了中国省际经济差距状况及来源。研究发现:总的来看,中国省区劳均产出与要素投入均呈现空间σ趋异倾向,而全要素生产率则在呈现空间σ收敛后趋于稳定。进一步利用VAR模型及脉冲响应函数分析发现,在较长时期内,要素投入总体上扩大了中国省际经济差距,而全要素生产率则有助于缩小省际经济差距,这主要是由要素投入的经济拉动滞后效应和技术的扩散效应所致。     

Consumer Search Behavior in the Changing Credit Card Market

Many recent articles have identified behavioral disturbances in vector autoregressions by imposing restrictions on the long-run effects of shocks. This article demonstrates that this approach will be unreliable unless the underlying economy satisfies three types of strong restrictions. Although many aspects of these issues have been raised before, this article draws out and illustrates the implications for inferences under the long-run scheme. Furthermore, it provides strategies for dealing with the problems.     

中国国际收支经常账户与资本金融账户关系研究：1981-2007

 本文针对中国国际收支多年来“双顺差”的事实,利用1981～2007年度中国国际收支时序数据和我们认为与之相关的最重要的两个宏观经济变量：中国的对外开放度指标和实际GDP的增长率数据,构建了多组向量自回归（VAR）模型,通过格兰杰因果检验、脉冲响应函数和方差分解等计量方法,实证检验了中国国际收支经常账户及其子账户与资本金融账户之间,各账户与宏观经济变量之间的因果关系、动态冲击响应和变量间影响的相对重要性。