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.     

Discussion: The Recent Reappearance of the Homotheticity Restriction on Preferences

A model linking final demand (sales) with intermediate manufactures is estimated and simulated over the course of a business cycle. The results show that prices fluctuate more for early-stage manufactures than for later-stage products at the manufacturing and retail level. This is found to be largely due to inventory and unfilled orders fluctuations, particularly those of inventories of primary manufactured products held by finished goods manufacturers. Since these inventory fluctuations lead the cycle in final demand, prices of primary manufactures lead business activity. Another result is that inventories of materials and supplies, including those of retailers for resale, exhibit more instability than producers' inventories of finished goods.     

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.     

协整平滑转移回归中的线性检验 ——基于完全修正最小二乘法的扩展

 在解释变量内生条件下,Choi,Saikkonen(2004)使用动态最小二乘法估计协整平滑转移回归模型,并基于动态最小二乘的估计结果构造 统计量检验协整向量的非线性。本文系统解析了 的构造并指出其不足,针对这一不足,本文将动态最小二乘法扩展为完全修正的最小二乘法,并进而基于完全修正的最小二乘法估计结果构造 统计量检验协整向量的非线性。本文的仿真试验表明,在有限样本下, 与 的检验势没有显著差异,但 的水平扭曲小于 。     

The selection of zero-non-zero patterned cointegrating vectors in error-correction modelling

An effective and efficient search algorithm has been developed to select from an 1(1) system zero-non-zero patterned cointegrating and loading vectors in a subset VECM, B ^q (l)y(t-1) + B ^q-1 (L)Ay(t) = ?( t ) , where the long term impact matrix Bq(l) contains zero entries. The algorithm can be applied to higher order integrated systems. The Finnish money-output model presented by Johansen and Juselius (1990) and the United States balanced growth model presented by King, Plosser, Stock and Watson (1991) are used to demonstrate the usefulness of this algorithm in examining the cointegrating relationships in vector time series.     

The Decline in U.S. Output Volatility

Explanations for the decline in U.S. output volatility since the mid-1980s include: “better policy,” “good luck,” and technological change. Our multiple-break estimates suggest that reductions in volatility since the mid-1980s extend not only to manufacturing inventories, but also to sales. This finding, along with a concentration of the reduction in the volatility of inventories in materials and supplies and the lack of a significant break in the inventory–sales covariance, imply that new inventory technology cannot account for most of the decline in output volatility.     

Exploring the Impact of Multivariate GARCH Innovations on Hypothesis Testing for Cointegrating Vectors

This article investigates the impact of multivariate generalized autoregressive conditional heteroskedastic (GARCH) errors on hypothesis testing for cointegrating vectors. The study reviews a cointegrated vector autoregressive model incorporating multivariate GARCH innovations and a regularity condition required for valid asymptotic inferences. Monte Carlo experiments are then conducted on a test statistic for a hypothesis on the cointegrating vectors. The experiments demonstrate that the regularity condition plays a critical role in rendering the hypothesis testing operational. It is also shown that Bartlett-type correction and wild bootstrap are useful in improving the small-sample size and power performance of the test statistic of interest.     

The Size and Power of Bootstrap and Bartlett-Corrected Tests of Hypotheses on the Cointegrating Vectors

In this paper we compare Bartlett-corrected, bootstrap, and fast double bootstrap tests on maximum likelihood estimates of cointegration parameters. The key result is that both the bootstrap and the Bartlett-corrected tests must be based on the unrestricted estimates of the cointegrating vectors: procedures based on the restricted estimates have almost no power. The small sample size bias of the asymptotic test appears so severe as to advise strongly against its use with the sample sizes commonly available; the fast double bootstrap test minimizes size bias, while the Bartlett-corrected test is somehow more powerful.     

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.     

Comments

In this paper we compare Bartlett-corrected, bootstrap, and fast double bootstrap tests on maximum likelihood estimates of cointegration parameters. The key result is that both the bootstrap and the Bartlett-corrected tests must be based on the unrestricted estimates of the cointegrating vectors: procedures based on the restricted estimates have almost no power. The small sample size bias of the asymptotic test appears so severe as to advise strongly against its use with the sample sizes commonly available; the fast double bootstrap test minimizes size bias, while the Bartlett-corrected test is somehow more powerful.     

Parameter estimation and inference with spatial lags and cointegration

This article studies dynamic panel data models in which the long run outcome for a particular cross-section is affected by a weighted average of the outcomes in the other cross-sections. We show that imposing such a structure implies a model with several cointegrating relationships that, unlike in the standard case, are nonlinear in the coe?cients to be estimated. Assuming that the weights are exogenously given, we extend the dynamic ordinary least squares methodology and provide a dynamic two-stage least squares estimator. We derive the large sample properties of our proposed estimator under a set of low-level assumptions. Then our methodology is applied to US financial market data, which consist of credit default swap spreads, as well as firm-specific and industry data. We construct the economic space using a “closeness” measure for firms based on input–output matrices. Our estimates show that this particular form of spatial correlation of credit default swap spreads is substantial and highly significant.     

Pitfalls in Estimating Cointegrating Vector when Cointegration Relationship has Nonlinear Adjustment

This article investigates the properties of the estimators of the cointegrating vector when the cointegration error has a nonlinear adjustment. We investigate the properties of three estimators, namely, ordinary least squares (OLS), dynamic OLS (DOLS), and autoregressive distributed lag (ADL) models. Monte Carlo simulation results demonstrate that although all the estimators have consistency under cointegration with a nonlinear adjustment, they suffer from severe size distortions for the t-statistics of the cointegrating vector when the cointegration error has a highly persistent nonlinear adjustment and endogeneity. The results imply that the use of DOLS and ADL for cointegration with nonlinear adjustment cannot sufficiently improve the estimates and size performances.     

Aliternative methods for estimating long-run responses with applications to australian import demand

This paper compares and contrasts a number of single-equation and systems estimators of long-run responses with application to a three-variable import demand model. Two variants of Box and Tiao's (1977) canonical estimator are developed and associated tests for the number of cointegrating vectors are introduced. A simulation study indicates that, while both Box-Tiao estimators have empirical distributions with fatter tails than the normal, there is evidence that the incidence of extreme values is even greater with Johansen's (1988) ML procedure.     

Aliternative methods for estimating long-run responses with applications to australian import demand

This paper compares and contrasts a number of single-equation and systems estimators of long-run responses with application to a three-variable import demand model. Two variants of Box and Tiao's (1977) canonical estimator are developed and associated tests for the number of cointegrating vectors are introduced. A simulation study indicates that, while both Box-Tiao estimators have empirical distributions with fatter tails than the normal, there is evidence that the incidence of extreme values is even greater with Johansen's (1988) ML procedure.     

The effect of non-normal disturbances and conditional heteroskedasticity on multiple cointegration tests

!t is well-known that Johansen's multiple cointegration tests' results and those of Johansen and Juselius' tests for restricrions on cointegrating vectors and their weights have far-reaching implications for economic modelling and analysis. Therefore, it is important to ensure that the tests have desirable finite sample properties. Although the statistics are derived under Gaussian distribution,the asympotic results are derived under a much wider class of distributions. Using simulation, this paper investigates the effect of non-normal disturbances on these tests in finite samples. Further, ARCH/GARCH type conditional heteroskedasticity is present in many economic and financial time series. This paper examines the finite properties of the tests when the error term follows ARCH/GARCH type processes. From the evidence, it appears that researchers should not be overly concerned by the possibility of small departures from non-normality when using Johansen's suggested techniques even in finite samples. ARCH and GARCH effects may be more problematic, however. In particular it becomes more important ro test whether the restriction implicit in the integrated (or near-integrated) ARCH-type Drocess actually holds in time series for the application of the cointegraiion rank tests and the test for restrictions on cointegrating weights. The tests for restrictions on cointegrating vectors apper to be robust for non-normal errors and for all ARCH and GARCH type processes considered.     

Inflation and the Divergence of Relative Prices: Evidence from a Cointegration Analysis

The relation between inflation and RPV plays a prominent role in explaining the costs of inflation. This study investigates whether the CPI subcategories drift apart more over a period of high inflation rates than during one of low inflation. The wider dispersion of the subcategories is reflected in an increasing number of common stochastic trends in the system of sub price indices. The results for US data as well as for cross-country comparisons indicate that the influence of inflation on the dispersion of relative prices cannot be revealed by counting cointegrating relations. Thus, the number of stochastic trends or cointegrating relations is not a reliable indicator for the distorting effect of inflation on the dispersion of relative prices.     

The selection of zero-non-zero patterned cointegrating vectors in error-correction modelling

An effective and efficient search algorithm has been developed to select from an 1(1) system zero-non-zero patterned cointegrating and loading vectors in a subset VECM, B^q(l)y(t-1) + B^q-1 (L)Ay(t) = ε( t ) , where the long term impact matrix Bq(l) contains zero entries. The algorithm can be applied to higher order integrated systems. The Finnish money-output model presented by Johansen and Juselius (1990) and the United States balanced growth model presented by King, Plosser, Stock and Watson (1991) are used to demonstrate the usefulness of this algorithm in examining the cointegrating relationships in vector time series.     

Bayesian Inference in Cointegrated I (2) Systems: A Generalization of the Triangular Model

This paper generalizes the cointegrating model of Phillips (1991) to allow for I (0), I (1) and I (2) processes. The model has a simple form that permits a wider range of I (2) processes than are usually considered, including a more flexible form of polynomial cointegration. Further, the specification relaxes restrictions identified by Phillips (1991) on the I (1) and I (2) cointegrating vectors and restrictions on how the stochastic trends enter the system. To date there has been little work on Bayesian I (2) analysis and so this paper attempts to address this gap in the literature. A method of Bayesian inference in potentially I (2) processes is presented with application to Australian money demand using a Jeffreys prior and a shrinkage prior.     

Bayesian Spatial Modeling of Housing Prices Subject to a Localized Externality

This work proposes a non stationary random field model to describe the spatial variability of housing prices that are affected by a localized externality. The model allows for the effect of the localized externality on house prices to be represented in the mean function and/or the covariance function of the random field. The correlation function of the proposed model is a mixture of an isotropic correlation function and a correlation function that depends on the distances between home sales and the localized externality. The model is fit using a Bayesian approach via a Markov chain Monte Carlo algorithm. A dataset of 437 single family home sales during 2001 in the city of Cedar Falls, Iowa, is used to illustrate the model.     

A Bayesian Approach to Modeling Time-Varying Cointegration and Cointegrating Rank

A multivariate model that allows for both a time-varying cointegrating matrix and time-varying cointegrating rank is presented. The model addresses the issue that, in real data, the validity of a constant cointegrating relationship may be questionable. The model nests the submodels implied by alternative cointegrating matrix ranks and allows for transitions between stationarity and nonstationarity, and cointegrating and noncointegrating relationships in accordance with the observed behavior of the data. A Bayesian test of cointegration is also developed. The model is used to assess the validity of the Fisher effect and is also applied to equity market data.