http://repub.eur.nl/res/aut/2613/ List of Publications en http://repub.eur.nl/static-eur/img/logo.png http://repub.eur.nl http://repub.eur.nl/res/pub/11132/ 2007-05-01T00:00:00Z We propose a natural conjugate prior for the instrumental variables regression model. The prior is a natural conjugate one since the marginal prior and posterior of the structural parameter have the same functional expressions which directly reveal the update from prior to posterior. The Jeffreys prior results from a specific setting of the prior parameters and results in a marginal posterior of the structural parameter that has an identical functional form as the sampling density of the limited information maximum likelihood estimator. We construct informative priors for the Angrist–Krueger [1991. Does compulsory school attendance affect schooling and earnings? Quarterly Journal of Economics 106, 979–1014] data and show that the marginal posterior of the return on education in the US coincides with the marginal posterior from the Southern region when we use the Jeffreys prior. This result occurs since the instruments are the strongest in the Southern region and the posterior using the Jeffreys prior, identical to maximum likelihood, focusses on the strongest available instruments. We construct informative priors for the other regions that make their posteriors of the return on education similar to that of the US and the Southern region. These priors show the amount of prior information needed to obtain comparable results for all regions. http://repub.eur.nl/res/pub/13216/ 2006-07-01T00:00:00Z We propose a novel statistic to test the rank of a matrix. The rank statistic overcomes deficiencies of existing rank statistics, like: a Kronecker covariance matrix for the canonical correlation rank statistic of Anderson [Annals of Mathematical Statistics (1951), 22, 327–351] sensitivity to the ordering of the variables for the LDU rank statistic of Cragg and Donald [Journal of the American Statistical Association (1996), 91, 1301–1309] and Gill and Lewbel [Journal of the American Statistical Association (1992), 87, 766–776] a limiting distribution that is not a standard chi-squared distribution for the rank statistic of Robin and Smith [Econometric Theory (2000), 16, 151–175] usage of numerical optimization for the objective function statistic of Cragg and Donald [Journal of Econometrics (1997), 76, 223–250] and ignoring the non-negativity restriction on the singular values in Ratsimalahelo [2002, Rank test based on matrix perturbation theory. Unpublished working paper, U.F.R. Science Economique, University de Franche-Comté]. In the non-stationary cointegration case, the limiting distribution of the new rank statistic is identical to that of the Johansen trace statistic. http://repub.eur.nl/res/pub/7247/ 2006-01-24T00:00:00Z We propose a natural conjugate prior for the instrumental variables regression model. The prior is a natural conjugate one since the marginal prior and posterior of the structural parameter have the same functional expressions which directly reveal the update from prior to posterior. The Jeffreys prior results from a specific setting of the prior parameters and results in a marginal posterior of the structural parameter that has an identical functional form as the sampling density of the limited information maximum likelihood estimator. We construct informative priors for the Angrist-Krueger (1991) data and show that the marginal posterior of the return on education in the US coincides with the marginal posterior from the Southern region when we use the Jeffreys prior. This result occurs since the instruments are the strongest in the Southern region and the posterior using the Jeffreys prior, identical to maximum likelihood, focusses on the strongest available instruments. We construct informative priors for the other regions that make their posteriors of the return on education similar to that of the US and the Southern region. These priors show the amount of prior information needed to obtain comparable results for all regions. http://repub.eur.nl/res/pub/1681/ 2003-02-17T00:00:00Z We propose a novel statistic to test the rank of a matrix. The rank statistic overcomes deficiencies of existing rank statistics, like: necessity of a Kronecker covariance matrix for the canonical correlation rank statistic of Anderson (1951), sensitivity to the ordering of the variables for the LDU rank statistic of Cragg and Donald (1996) and Gill and Lewbel (1992), a limiting distribution that is not a standard chi-squared distribution for the rank statistic of Robin and Smith (2000) and usage of numerical optimization for the objective function statistic of Cragg and Donald (1997). The new rank statistic consists of a quadratic form of a (orthogonal) transformation of the smallest singular values of a unrestricted estimate of the matrix of interest. The quadratic form is taken with respect to the inverse of a unrestricted covariance matrix that can be estimated using a heteroscedasticity autocorrelation consistent estimator. The rank statistic has a standard chi squared limiting distribution. In case of a Kronecker covariance matrix, the rank statistic simplifies to the canonical correlation rank statistic. In the non-stationary cointegration case, the limiting distribution of the rank statistic is identical to that of the Johansen trace statistic. We apply the rank statistic to test for the rank of a matrix that governs the identification of the parameters in the stochastic discount factor model of Jagannathan and Wang (1996). The rank statistic shows that non-identification of the parameters can not be rejected. We further use the stochastic discount factor model to illustrate the validity of the limiting distribution and to conduct a power comparison. http://repub.eur.nl/res/pub/13232/ 2002-12-01T00:00:00Z Cointegration occurs when the long-run multiplier matrix of a vector autoregressive model exhibits rank reduction. Using a singular value decomposition of the unrestricted long-run multiplier matrix, we construct a parameter that reflects the presence of rank reduction. Priors and posteriors of the parameters of the cointegration model follow from conditional priors and posteriors of the unrestricted long-run multiplier matrix given that the parameter that reflects rank reduction is equal to zero. This idea leads to a complete Bayesian framework for cointegration analysis. It includes prior specification, simulation schemes for obtaining posterior distributions and determination of the cointegration rank via Bayes factors. We apply the proposed Bayesian cointegration analysis to the Danish data of Johansen and Juselius (Oxford Bull. Econom. Stat. 52 (1990) 169). http://repub.eur.nl/res/pub/7691/ 2000-04-19T00:00:00Z We propose a novel Bayesian test under a (noninformative) Jeffreys' prior specification. We check whether the fixed scalar value of the so- called Bayesian Score Statistic (BSS) under the null hypothesis is a plausible realization from its known and standardized distribution under the alternative. Unlike highest posterior density regions the BSS is invariant to reparameterizations. The BSS equals the posterior expectation of the classical score statistic and it provides an exact test procedure, whereas classical tests often rely on asymptotic results. Since the statistic is evaluated under the null hypothesis it provides the Bayesian counterpart of diagnostic checking. This result extends the similarity of classical sampling densities of maximum likelihood estimators and Bayesian posterior distributions based on Jeffreys' priors, towards score statistics. We illustrate the BSS as a diagnostic to test for misspecification in linear and cointegration models. http://repub.eur.nl/res/pub/18192/ 2000-01-01T00:00:00Z We propose a novel Bayesian test under a (noninformative) Jeffreys’ prior specifica- tion. We check whether the fixed scalar value of the so-called Bayesian Score Statistic (BSS) under the null hypothesis is a plausible realization from its known and standard- ized distribution under the alternative. Unlike highest posterior density regions the BSS is invariant to reparameterizations. The BSS equals the posterior expectation of the classical score statistic and it provides an exact test procedure, whereas classical tests often rely on asymptotic results. Since the statistic is evaluated under the null hypothe- sis it provides the Bayesian counterpart of diagnostic checking. This result extends the similarity of classical sampling densities of maximum likelihood estimators and Bayesian posterior distributions based on Jeffreys’ priors, towards score statistics. We illustrate the BSS as a diagnostic to test for misspecification in linear and cointegration models. http://repub.eur.nl/res/pub/7718/ 1999-07-28T00:00:00Z We propose in this paper a likelihood-based framework for cointegration analysis in panels of a fixed number of vector error correction models. Maximum likelihood estimators of the cointegrating vectors are constructed using iterated Generalized Method of Moments estimators. Using these estimators we construct likelihood ratio statistics to test for a common cointegration rank across the individual vector error correction models, both with heterogeneous and homogeneous cointegrating vectors. The corresponding limiting distributions are a summation of the limiting behavior of Johansen (1991) trace statistics. We also incorporate both unrestricted and restricted deterministic components which are either homogeneous or heterogeneous. The proposed framework is applied on a data set of exchange rates and appropriate monetary fundamentals. The test results show strong evidence for the validity of the monetary exchange rate model within a panel of vector error correction models for three major European countries, whereas the results based on individual vector error correction models for each of these countries separately are less supportive. http://repub.eur.nl/res/pub/1586/ 1999-03-31T00:00:00Z We present a new framework for the joint estimation of the default-free government term structure and corporate credit spread curves. By using a high-quality data set of German mark denominated bonds, we show that this yields more realistic spreads than conventionally obtained spread curves that result from subtracting independently estimated government and corporate term structures. The estimated spread curves are now smooth functions of time to maturity, as opposed to the twisting curves one gets from the traditional method, and are less sensitive to model specifications. Moreover, the implied corporate term structures have tighter confidence intervals. http://repub.eur.nl/res/pub/11328/ 1999-01-01T00:00:00Z The effect which the oil price time series has on the long run properties of Vector AutoRegressive (VAR) models for price levels and import demand is investigated. As the oil price variable is assumed to be weakly exogenous for the long run parameters, a cointegration testing procedure allowing for weakly exogenous variables is developed using a LU decomposition of the long run multiplier matrix. The likelihood based cointegration test statistics, Wald, Likelihood Ratio and Lagrange Multiplier, are constructed and their limiting distributions derived. Using these tests, we find that incorporating the oil price in a model for the domestic or import price level of seven industrialized countries decreases the long run memory of the inflation rate. Second, we find that the results for import demand can be classified with respect to the oil importing or exporting status of the specific country. The result for Japan is typical as its import price is not influenced by GNP in the long run, which is the case for all other countries. http://repub.eur.nl/res/pub/1561/ 1999-01-01T00:00:00Z We consider representation, estimation and inference on cointegration in a (PVAR). We show that cointegration amounts to a restriction on a product of parameter matrices. We therefore use GMM to construct estimators of the long-run (cointegration) parameters and to obtain test statistics for cointegration. We show that the limiting distributions of the GMM estimators and the corresponding test statistics in a PVAR are identical to those of the maximum likelihood cointegration estimators and test statistics in standard nonperiodic VAR models. http://repub.eur.nl/res/pub/1524/ 1998-12-31T00:00:00Z Statistical inference in nested linear models that result from linear restrictions on the parameters of encompassing linear models can be considered to result from the conditional distribution under the encompassing model. We extend this reasoning to nested models that result from general (nonlinear) restrictions by defining sufficient conditions that, when satisfied by the random variables and the restrictions, ensure the existence of an unique expression of the conditional density. Statistical inference in these nested models can then be considered to result from such a conditional density. This novel manner of conducting statistical analyzes leads both to some new results and allows one to obtain some already known results in a different manner. In Bayesian statistics, the conditional densities show how to construct specific classes of priors for the parameters of nested models, priors on the parameters of encompassing models that imply an already specified prior on the parameters of the nested model, Bayes factors using (generalized) Savage-Dickey density ratios and Bayesian score statistics. In classical statistical analysis, the conditional densities offer an alternative approach for constructing small sample and limiting distributions of maximum likelihood estimators and classical score statistics. http://repub.eur.nl/res/pub/1529/ 1998-11-26T00:00:00Z We construct limiting and small sample distributions of maximum likelihood estimators (mle) from the property that they satisfy the first order condition (foc). The foc relates the mle of the analyzed model to the mle of an encompassing model and shows that the mle of the analyzed model is a realization from the limiting or small sample distribution of the mle of the encompassing model given that the foc holds. We can thus use the unique conditional (limiting or small sample) density of the mle of the encompassing model given that the foc holds to construct the limiting or small sample density/distribution of the mle of the analyzed model. To proof the validity of this approach and thus of the concept of an unique conditional density, we use it to construct the small sample and limiting distribution of the limited information mle and show that they are identical to resp. the sampling density and the expression discussed elsewhere in the literature. We analyze the further and relate it to existing expressions and show its limiting behavior in case of weak and strong instruments. http://repub.eur.nl/res/pub/1540/ 1998-11-26T00:00:00Z We establish the relationships between certain Bayesian and classical approaches to instrumental variable regression. We determine the form of priors that lead to posteriors for structural parameters that have similar properties as classical 2SLS and LIML and in doing so provide some new insight to the small sample behavior of Bayesian and classical procedures in the limited information simultaneous equations model. Our approach is motivated by the relationship between Bayesian and classical procedures in linear regression models; i.e., Bayesian analysis with a diffuse prior leads to posteriors that are identical in form to the finite sample density of classical least squares estimators. We use the fact that the instrumental variables regression model can be obtained from a reduced rank restriction on a multivariate linear model to determine the priors that give rise to posteriors that have properties similar to classical 2SLS and LIML. As a by-product of this approach we provide a novel way to determine the exact finite sample density of the LIML estimator and the prior that corresponds with classical LIML. We show that the traditional Dreze (1976) and a new Bayesian Two Stage approach are similar to 2SLS whereas the approach based on the Jeffreys' prior corresponds to LIML. http://repub.eur.nl/res/pub/1551/ 1998-07-02T00:00:00Z Cointegration occurs when the long run multiplier of a vector autoregressive model exhibits rank reduction. Priors and posteriors of the parameters of the cointegration model are therefore proportional to priors and posteriors of the long run multiplier given that it has reduced rank. Rank reduction of the long run multiplier is modelled using a decomposition resulting from its singular value decomposition. It specifies the long run multiplier matrix as the sum of a matrix that equals the product of the adjustment parameters and the cointegrating vectors, i.e. the cointegration specification, and a matrix that models the deviation from cointegration. Priors and posteriors for the parameters of the cointegration model are obtained by restricting the latter matrix to zero in the prior and posterior of the unrestricted long run multiplier. The special decomposition of the long run multiplier results in unique posterior densities. This theory leads to a complete Bayesian framework for cointegration analysis. It includes prior specification, simulation schemes for obtaining posterior distributions and determination of the cointegration rank via Bayes factors. We illustrate the analysis with several simulated series, the UK data of Hendry and Doornik (1994) and the Danish data of Johansen and Juselius (1990). http://repub.eur.nl/res/pub/7822/ 1997-01-23T00:00:00Z Parameters in AutoRegressive Moving Average (ARMA) models are locally nonidentified, due to the problem of root cancellation. Parameters can be constructed which represent this identification problem. We argue that ARMA parameters should be analyzed conditional on these identifying parameters. Priors exploiting this feature result in regular posteriors, while priors which neglect it result in posteriori favor of nonidentified parameter values. By considering the implicit AR representation of an ARMA model a prior with the desired proporties is obtained. The implicit AR representation also allows to construct easily implemented algorithms to analyse ARMA parameters. As a byproduct, posteriors odds ratios can be computed to compare (nonnested) parsimonious ARMA models. The procedures are applied to two datasets, the (extended) Nelson-Plosser data and monthly observations of US 3-month and 10 year interest rates. For approximately 50% of the series in these two datasets an ARMA model is favored above an AR model. http://repub.eur.nl/res/pub/1410/ 1997-01-01T00:00:00Z Generalized Method of Moments (GMM) Estimators are derived for Reduced Rank Regression Models, the Error Correction Cointegration Model (ECCM) and the Incomplete Simultaneous Equations Model (INSEM). The GMM (2SLS) estimators of the cointegrating vector in the ECCM are shown to have normal limiting distributions. Tests for the number of unit roots can be constructed straightforwardly and have Dickey-Fuller type limiting distributions. Two extensions of the ECCM, which are important in practice, are analyzed. First, cointegration estimators and tests allowing for structural shifts in the variance (heteroscedasticity) of the series are derived and analyzed using a Generalized Least Squares Estimator. Second, cointegrating vector estimators and tests are derived which allow for structural breaks in the cointegrating vector and/or multiplicator. The resulting cointegrating vectors estimators have again normal limiting distributions while the cointegration tests have limiting distributions which differ from the standard Dickey-Fuller type. http://repub.eur.nl/res/pub/1414/ 1997-01-01T00:00:00Z Diffuse priors lead to pathological posterior behavior when used in Bayesian analyses of Simultaneous Equation Models (SEMs). This results from the local nonidentification of certain parameters in SEMs. When this, a priori known, feature is not captured appropriately, an a posteriori favor for certain specific parameter values results which is not the consequence of strong data information but of local nonidentification. We show that a proper consistent Bayesian analysis of a SEM explicitly has to consider the reduced form of the SEM as a standard linear model on which nonlinear (reduced rank) restrictions are imposed, which result from a singular value decomposition. The priors/posteriors of the parameters of the SEM are therefore proportional to the priors/posteriors of the parameters of the linear model under the condition that the restrictions hold. This leads to a framework for constructing priors and posteriors for the parameters of SEMs. The framework is used to construct priors and posteriors for one, two and three structural equation SEMs. These examples jointly with a theorem, which states that the reduced forms of SEMs accord with sets of reduced rank restrictions on standard linear models, show how Bayesian analyses of generally specified SEMs are conducted. http://repub.eur.nl/res/pub/1418/ 1997-01-01T00:00:00Z The effect which the oil price time series has on the long run properties of Vector AutoRegressive (VAR) models for price levels and import demand is investigated. As the oil price variable is assumed to be weakly exogenous for the long run parameters, a cointegration testing procedure allowing for weakly exogenous variables is developed using a LU\\ decomposition of the long run multiplier matrix. The likelihood based cointegration test statistics, Wald, Likelihood Ratio and Lagrange Multiplier, are constructed and their limiting distributions derived. Using these tests, we find that incorporating the oil price in a model for the domestic or import price level of seven industrialized countries decreases the long run memory of the inflation rate. Second, we find that the results for import demand can be classified with respect to the oil importing or exporting status of the specific country. The result for Japan is typical as its import price is not influenced by gnp in the long run, which is the case for all other countries. http://repub.eur.nl/res/pub/2095/ 1996-06-01T00:00:00Z In this paper we compare two univariate time series models, i.e. one with and one without an imposed unit root, in a forecasting experiment for the fourteen annually observed US data analyzed by Nelson and Plosser (1982, Journal of Monetary Economics 10, 139–162). Our main result is that the unit root model is regularly preferred. This result holds for a variety of sample sizes and forecast horizons as well as for one-step and multi-step ahead forecasts. http://repub.eur.nl/res/pub/1396/ 1996-01-01T00:00:00Z Many common statistical models can be specified as linear models with restrictions imposed on the parameters. A large amount of these models impose restrictions which do not allow for the analytical construction of the probability density function (pdf) of the parameters given the restrictions. This is often implicitly assumed which leads to an inconsistency as the pdf of the parameters of the linear specification under the imposed restrictions is then not nested within the assumed pdf of the unrestricted linear specification. The paper shows how these restrictions need to be incorporated by constructing the pdfs incorparating the restrictions and algorithms to sample from these pdfs. We show how these methods are applied to some common statistical models, i.e. ARMA, cointegration and simultaneous equation models. http://repub.eur.nl/res/pub/1398/ 1996-01-01T00:00:00Z Using the standard linear model as a base, a unified theory of Bayesian Analyses of Cointegration Models is constructed. This is achieved by defining (natural conjugate) priors in the linear model and using the implied priors for the cointegration model. Using these priors, posterior results for the cointegration model are obtained using a Metropolis-Hasting sampler. To compare the cointegration models mutually and with the vector autoregressive model under stationarity, we use two strategies. The first strategy uses the Bayesian interpretation of a Lagrange Multiplier statistic. The second strategy compares the models using prior and posterior odds ratios. The latter enables us to compute prior and posterior distributions over the cointegration rank and shows close resemblance with the posterior information criterium from Phillips and Ploberger (1996). To show the applicability of the derived theory, the constructed procedures are applied to data from Johansen and Juselius (1990) and a few simulated data sets. http://repub.eur.nl/res/pub/1363/ 1995-01-01T00:00:00Z Parameters in AutoRegressive Moving Average (ARMA) models are locally nonidentified, due to the problem of root cancellation. Parameters can be constructed which represent this identification problem. We argue that ARMA parameters should be analyzed conditional on these identifying parameters. Priors exploiting this feature result in regular posteriors, while priors which neglect it result in posteriori favor of nonidentified parameter values. By considering the implicit AR representation of an ARMA model a prior with the desired proporties is obtained. The implicit AR representation also allows to construct easily implemented algorithms to analyze ARMA parameters. As a byproduct, posteriors odds ratios can be computed to compare (nonnested) parsimonious ARMA models. The procedures are applied to two datasets, the (extended) Nelson-Plosser data and monthly observations of US 3-month and 10 year interest rates. For approximately 50% of the series in these two datasets an ARMA model is favored above an AR model. http://repub.eur.nl/res/pub/11296/ 1994-01-01T00:00:00Z Abstract An error correction model is specified having only exact identified parameters, some of which reflect a possible departure from a cointegration model. Wald, likelihood ratio, and Lagrange multiplier statistics are derived to test for the significance of these parameters. The construction of the Wald statistic only involves linear regression, and under certain conditions the limiting distribution of the Wald statistic differs from the limiting distributions of the likelihood ratio and Lagrange multiplier statistics. A special ordering of the variables is recommended so that equal limiting distributions of the three different test statistics are obtained. The applicability of the derived testing procedures is illustrated using real demand for money, real GNP, and bond and deposit interest rates from Denmark. http://repub.eur.nl/res/pub/11248/ 1993-01-01T00:00:00Z First, the non-stationarity properties of the conditional variances in the GARCH(1,1) model are analysed using the concept of infinite persistence of shocks. Given a time sequence of probabilities for increasing/decreasing conditional variances, a theoretical formula for quasi-strict non-stationarity is defined. The resulting conditions for the GARCH(1,1) model are shown to differ from the weak stationarity conditions mainly used in the literature. Bayesian statistical analysis using Monte Carlo integration is applied to analyse both stationarity concepts for the conditional variances of the US 3-month treasury bill rate. Interest rates are known for their weakly non-stationary conditional variances but, using a quasi-strict stationarity measure, it is shown that the conditional variances are likely to be stationary. Second, the level of the treasury bill rate is analysed for non-stationarity using Bayesian unit root methods. The disturbances of the GARCH model for the treasury bill rate are t-distributed. It is shown that the unit root parameter is negatively correlated with the degrees-of-freedom parameter. Imposing normally distributed disturbances leads therefore to underestimation of the non-stationarity in the level of the treasury bill rate.