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

This paper generalizes the cointegrating model of Phillips (1991 Phillips , P. C. B. ( 1991 ). Optimal inference in cointegrated systems . Econometrica 59 : 283 – 306 .[Crossref], [Web of Science ®] , [Google Scholar]) 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 Phillips , P. C. B. ( 1991 ). Optimal inference in cointegrated systems . Econometrica 59 : 283 – 306 .[Crossref], [Web of Science ®] , [Google Scholar]) 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.     

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.     

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.     

An approach to evaluating sensitivity in Bayesian regression analyses

This paper presents a method for assessing the sensitivity of predictions in Bayesian regression analyses. In parametric Bayesian analyses there is a family s₀ of regression functions, parametrized by a finite-dimensional vector B. The family s₀ is a subset of R, the set of all possible regression functions. A prior π₀ on B induces a prior on R. This paper assesses sensitivity by computing bounds on the predictive probability of a fixed set K over a class of priors, Γ, induced by a class of families of regression functions, Γ_s, and a class of priors, Γ_π. This paper is divided into three parts which (1) define Γ, (2) describe an algorithm for finding accurate bounds on predictive probabilities over Γ and (3) illustrate the method with two examples. It is found that sensitivity to the family of regression functions can be much more important than sensitivity to π₀.     

The flower intersection problem for Kirkman triple systems

The flower at a point x in a Steiner triple system is the set of all triples containing x. Denote by I_R^*[r] the set of all integers k such that there exists a pair of KTS(2r+1) having k+r triples in common, r of them being the triples of a common flower. In this article we determine the set I_R^*[r] for any positive integer r≡1 (mod 3) (only nine cases are left undecided for r=7,13,16,19), and establish that I_R^*[r]=J[r] for r≡1 (mod 3) and r22 where J[r]={0,1,…,2r(r−1)/3−6,2r(r−1)/3−4,2r(r−1)/3}.     

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.     

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.     

A SCORE TEST FOR SEASONAL FRACTIONAL INTEGRATION AND COINTEGRATION

This paper develops a time domain score statistic for testing fractional integration at zero and seasonal frequencies in quarterly time series models. Further, it introduces the notion of fractional cointegration at different frequencies between two seasonally integrated, I(1) series. In testing problems involving seasonal fractional cointegration, it is argued that the alternative hypothesis is one-sided for which the usual score test may not be appropriate. Therefore, based on ideas in Silvapulle and Silvapulle (1995), a one-sided score statistic is constructed. A simulation study finds that the score statistic generally has desirable size and power properties in moderately sized samples. The score test is applied to the quarterly Australian consumption function. The income and consumption series are found to be I(1) at zero and seasonal frequencies and these two series are not cointegrated at any frequency.     

Analysis of cointegrated models with measurement errors

We study the asymptotic properties of the reduced-rank estimator of error correction models of vector processes observed with measurement errors. Although it is well known that there is no asymptotic measurement error bias when predictor variables are integrated processes in regression models [Phillips BCB, Durlauf SN. Multiple time series regression with integrated processes. Rev Econom Stud. 1986;53:473–495], we systematically investigate the effects of the measurement errors (in the dependent variables as well as in the predictor variables) on the estimation of not only cointegrating vectors but also the speed of the adjustment matrix. Furthermore, we present the asymptotic properties of the estimators. We also obtain the asymptotic distribution of the likelihood ratio test for the cointegrating ranks. We investigate the effects of the measurement errors on estimation and test through a Monte Carlo simulation study.     

Identification of local clusters for count data: a model-based Moran's I test

We set out I_DR as a loglinear-model-based Moran's I test for Poisson count data that resembles the Moran's I residual test for Gaussian data. We evaluate its type I and type II error probabilities via simulations, and demonstrate its utility via a case study. When population sizes are heterogeneous, I_DR is effective in detecting local clusters by local association terms with an acceptable type I error probability. When used in conjunction with local spatial association terms in loglinear models, I_DR can also indicate the existence of first-order global cluster that can hardly be removed by local spatial association terms. In this situation, I_DR should not be directly applied for local cluster detection. In the case study of St. Louis homicides, we bridge loglinear model methods for parameter estimation to exploratory data analysis, so that a uniform association term can be defined with spatially varied contributions among spatial neighbors. The method makes use of exploratory tools such as Moran's I scatter plots and residual plots to evaluate the magnitude of deviance residuals, and it is effective to model the shape, the elevation and the magnitude of a local cluster in the model-based test.     

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.     

A Bayesian nonparametric estimator based on left censored data

Summary This paper introduces a Bayesian nonparametric estimator for an unknown distribution function based on left censored observations. Hjort (1990)/Lo (1993) introduced Bayesian nonparametric estimators derived from beta/beta-neutral processes which allow for right censoring. These processes are taken as priors from the class ofneutral to the right processes (Doksum, 1974). The Kaplan-Meier nonparametric product limit estimator can be obtained from these Bayesian nonparametric estimators in the limiting case of a vague prior. The present paper introduces what can be seen as the correspondingleft beta/beta-neutral process prior which allow for left censoring. The Bayesian nonparametyric estimator is obtained as in the corresponding product limit estimator based on left censored data.     

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.     

Testing normalization and overidentification of cointegrating vectors in vector autoregressive processes

This paper develops test procedures for testing the validity of general linear identifying restrictions imposed on cointegrating vectors in the context of a vector autoregressive model. In addition to overidentifying restrictions the considered restrictions may also involve normalizing restrictions. Tests for both types of restrictions are developed and their asymptotic properties are obtained. Under the null hypothesis tests for normalizing restrictions have an asymptotic "multivariate unit root distribution", similar to that obtained for the likelihood ratio test for cointegration, while tests for overidentifying restrictions have a standard chi-square limiting distribution. Since these two types of tests are asymptotically independent they are easy to cotnbine to an overall test for the spccifed identifying restrictions. An overall test of this kind can consistently reveal the failure of the identifying restrictions in a wider class of cases than previous tests which only test for overidentifying restrictions.     

Bayesian Identification of Multivariate Autoregressive Processes

Identification is one of the most important stages of a time series analysis. This paper develops a direct Bayesian technique to identify the order of multivariate autoregressive processes. By employing the conditional likelihood function and a matrix normal-Wishart prior density, or Jeffrey' vague prior, the proposed identification technique is based on deriving the exact posterior probability mass function of the model order in a convenient form. Then one may easily evaluate the posterior probabilities of the model order and choose the order that maximizes the posterior mass function to be the suitable order of the time series data being analyzed. Assuming the bivariate autoregressive processes, a numerical study, with different prior mass functions, is carried out to assess the efficiency of the proposed technique. The analysis of the numerical results supports the adequacy of the proposed technique in identifying the orders of multivariate autoregressive processes.     

Efficient Posterior Simulation for Cointegrated Models with Priors on the Cointegration Space

A message coming out of the recent Bayesian literature on cointegration is that it is important to elicit a prior on the space spanned by the cointegrating vectors (as opposed to a particular identified choice for these vectors). In previous work, such priors have been found to greatly complicate computation. In this article, we develop algorithms to carry out efficient posterior simulation in cointegration models. In particular, we develop a collapsed Gibbs sampling algorithm which can be used with just-identifed models and demonstrate that it has very large computational advantages relative to existing approaches. For over-identifed models, we develop a parameter-augmented Gibbs sampling algorithm and demonstrate that it also has attractive computational properties.     

Efficient sampling schemes for Bayesian MARS models with many predictors

Multivariate adaptive regression spline fitting or MARS (Friedman 1991) provides a useful methodology for flexible adaptive regression with many predictors. The MARS methodology produces an estimate of the mean response that is a linear combination of adaptively chosen basis functions. Recently, a Bayesian version of MARS has been proposed (Denison, Mallick and Smith 1998a, Holmes and Denison, 2002) combining the MARS methodology with the benefits of Bayesian methods for accounting for model uncertainty to achieve improvements in predictive performance. In implementation of the Bayesian MARS approach, Markov chain Monte Carlo methods are used for computations, in which at each iteration of the algorithm it is proposed to change the current model by either (a) Adding a basis function (birth step) (b) Deleting a basis function (death step) or (c) Altering an existing basis function (change step). In the algorithm of Denison, Mallick and Smith (1998a), when a birth step is proposed, the type of basis function is determined by simulation from the prior. This works well in problems with a small number of predictors, is simple to program, and leads to a simple form for Metropolis-Hastings acceptance probabilities. However, in problems with very large numbers of predictors where many of the predictors are useless it may be difficult to find interesting interactions with such an approach. In the original MARS algorithm of Friedman (1991) a heuristic is used of building up higher order interactions from lower order ones, which greatly reduces the complexity of the search for good basis functions to add to the model. While we do not exactly follow the intuition of the original MARS algorithm in this paper, we nevertheless suggest a similar idea in which the Metropolis-Hastings proposals of Denison, Mallick and Smith (1998a) are altered to allow dependence on the current model. Our modification allows more rapid identification and exploration of important interactions, especially in problems with very large numbers of predictor variables and many useless predictors. Performance of the algorithms is compared in simulation studies.     

Posterior Bayes factor analysis for an exponential regression model

In the exponential regression model, Bayesian inference concerning the non-linear regression parameter has proved extremely difficult. In particular, standard improper diffuse priors for the usual parameters lead to an improper posterior for the non-linear regression parameter. In a recent paper Ye and Berger (1991) applied the reference prior approach of Bernardo (1979) and Berger and Bernardo (1989) yielding a proper informative prior for . This prior depends on the values of the explanatory variable, goes to 0 as goes to 1, and depends on the specification of a hierarchical ordering of importance of the parameters.This paper explains the failure of the uniform prior to give a proper posterior: the reason is the appearance of the determinant of the information matrix in the posterior density for . We apply the posterior Bayes factor approach of Aitkin (1991) to this problem; in this approach we integrate out nuisance parameters with respect to their conditional posterior density given the parameter of interest. The resulting integrated likelihood for requires only the standard diffuse prior for all the parameters, and is unaffected by orderings of importance of the parameters. Computation of the likelihood for is extremely simple. The approach is applied to the three examples discussed by Berger and Ye and the likelihoods compared with their posterior densities.     

Refined Inference on Long Memory in Realized Volatility

There is an emerging consensus in empirical finance that realized volatility series typically display long range dependence with a memory parameter (d) around 0.4 (Andersen et al., 2001; Martens et al., 2004). The present article provides some illustrative analysis of how long memory may arise from the accumulative process underlying realized volatility. The article also uses results in Lieberman and Phillips (2004, 2005) to refine statistical inference about d by higher order theory. Standard asymptotic theory has an O(n^-1/2) error rate for error rejection probabilities, and the theory used here refines the approximation to an error rate of o(n^-1/2). The new formula is independent of unknown parameters, is simple to calculate and user-friendly. The method is applied to test whether the reported long memory parameter estimates of Andersen et al. (2001) and Martens et al. (2004) differ significantly from the lower boundary (d = 0.5) of nonstationary long memory, and generally confirms earlier findings.