On Bayesian analysis and computation for functions with monotonicity and curvature restrictions

Our goal is inference for shape-restricted functions. Our functional form consists of finite linear combinations of basis functions. Prior elicitation is difficult due to the irregular shape of the parameter space. We show how to elicit priors that are flexible, theoretically consistent, and proper. We demonstrate that uniform priors over coefficients imply priors over economically relevant quantities that are quite informative and give an example of a non-uniform prior that addresses this issue. We introduce simulation methods that meet challenges posed by the shape of the parameter space. We analyze data from a consumer demand experiment.     

A semiparametric stochastic volatility model

In this paper the correlation structure in the classical leverage stochastic volatility (SV) model is generalized based on a linear spline. In the new model the correlation between the return and volatility innovations is time varying and depends nonparametrically on the type of news arrived to the market. Theoretical properties of the proposed model are examined. The model estimation and comparison are conducted by Bayesian methods. The performance of the estimates are examined in simulations. The new model is fitted to daily and weekly US data and compared with the classical SV and GARCH models in terms of their in-sample and out-of-sample performances. Empirical results suggest evidence in favor of the proposed model. In particular, the new model finds strong evidence of time varying leverage effect in individual stocks when the classical model fails to identify the leverage effect.     

Model selection, estimation and forecasting in VAR models with short-run and long-run restrictions

We study the joint determination of the lag length, the dimension of the cointegrating space and the rank of the matrix of short-run parameters of a vector autoregressive (VAR) model using model selection criteria. We suggest a new two-step model selection procedure which is a hybrid of traditional criteria and criteria with data-dependant penalties and we prove its consistency. A Monte Carlo study explores the finite sample performance of this procedure and evaluates the forecasting accuracy of models selected by this procedure. Two empirical applications confirm the usefulness of the model selection procedure proposed here for forecasting.     

Bayesian point estimation of the cointegration space

A neglected aspect of the otherwise fairly well developed Bayesian analysis of cointegration is point estimation of the cointegration space. It is pointed out here that, due to the well known non-identification of the cointegration vectors, the parameter space is not Euclidean and the loss functions underlying the conventional Bayes estimators are therefore questionable. We present a Bayes estimator of the cointegration space which takes the curved geometry of the parameter space into account. This estimate has the interpretation of being the posterior mean cointegration space and is invariant to the order of the time series, a property not shared with many of the Bayes estimators in the cointegration literature. An overall measure of cointegration space uncertainty is also proposed. Australian interest rate data are used for illustration. A small simulation study shows that the new Bayes estimator compares favorably to the maximum likelihood estimator.     

A spatial model for multivariate lattice data

In this article, we develop Markov random field models for multivariate lattice data. Specific attention is given to building models that incorporate general forms of the spatial correlations and cross-correlations between variables at different sites. The methodology is applied to a problem in environmental equity. Using a Bayesian hierarchical model that is multivariate in form, we examine the racial distribution of residents of southern Louisiana in relation to the location of sites listed with the U.S. Environmental Protection Agency's Toxic Release Inventory.     

Semiparametric Bayesian inference in smooth coefficient models

We describe procedures for Bayesian estimation and testing in cross-sectional, panel data and nonlinear smooth coefficient models. The smooth coefficient model is a generalization of the partially linear or additive model wherein coefficients on linear explanatory variables are treated as unknown functions of an observable covariate. In the approach we describe, points on the regression lines are regarded as unknown parameters and priors are placed on differences between adjacent points to introduce the potential for smoothing the curves. The algorithms we describe are quite simple to implement—for example, estimation, testing and smoothing parameter selection can be carried out analytically in the cross-sectional smooth coefficient model.     

Bayesian inference in a time varying cointegration model

There are both theoretical and empirical reasons for believing that the parameters of macroeconomic models may vary over time. However, work with time-varying parameter models has largely involved vector autoregressions (VARs), ignoring cointegration. This is despite the fact that cointegration plays an important role in informing macroeconomists on a range of issues. In this paper, we develop a new time varying parameter model which permits cointegration. We use a specification which allows for the cointegrating space to evolve over time in a manner comparable to the random walk variation used with TVP–VARs. The properties of our approach are investigated before developing a method of posterior simulation. We use our methods in an empirical investigation involving the Fisher effect.     

The finite-sample effects of VAR dimensions on OLS bias,OLS variance,and minimum MSE estimators

Vector autoregressions (VARs) are important tools in time series analysis. However, relatively little is known about the finite-sample behaviour of parameter estimators. We address this issue, by investigating ordinary least squares (OLS) estimators given a data generating process that is a purely nonstationary first-order VAR. Specifically, we use Monte Carlo simulation and numerical optimisation to derive response surfaces for OLS bias and variance, in terms of VAR dimensions, given correct specification and several types of over-parameterisation of the model: we include a constant, and a constant and trend, and introduce excess lags. We then examine the correction factors that are required for the least squares estimator to attain the minimum mean squared error (MSE). Our results improve and extend one of the main finite-sample multivariate analytical bias results of Abadir, Hadri and Tzavalis [Abadir, K.M., Hadri, K., Tzavalis, E., 1999. The influence of VAR dimensions on estimator biases. Econometrica 67, 163–181], generalise the univariate variance and MSE findings of Abadir [Abadir, K.M., 1995. Unbiased estimation as a solution to testing for random walks. Economics Letters 47, 263–268] to the multivariate setting, and complement various asymptotic studies.     

Methods for computing marginal data densities from the Gibbs output

We introduce two estimators for estimating the Marginal Data Density (MDD) from the Gibbs output. Our methods are based on exploiting the analytical tractability condition, which requires that some parameter blocks can be analytically integrated out from the conditional posterior densities. This condition is satisfied by several widely used time series models. An empirical application to six-variate VAR models shows that the bias of a fully computational estimator is sufficiently large to distort the implied model rankings. One of the estimators is fast enough to make multiple computations of MDDs in densely parameterized models feasible.     

Bayesian inference in a sample selection model

This paper develops methods of Bayesian inference in a sample selection model. The main feature of this model is that the outcome variable is only partially observed. We first present a Gibbs sampling algorithm for a model in which the selection and outcome errors are normally distributed. The algorithm is then extended to analyze models that are characterized by nonnormality. Specifically, we use a Dirichlet process prior and model the distribution of the unobservables as a mixture of normal distributions with a random number of components. The posterior distribution in this model can simultaneously detect the presence of selection effects and departures from normality. Our methods are illustrated using some simulated data and an abstract from the RAND health insurance experiment.     

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.     

Bootstrap testing for the null of no cointegration in a threshold vector error correction model

We develop a test for the linear no cointegration null hypothesis in a threshold vector error correction model. We adopt a sup-Wald type test and derive its null asymptotic distribution. A residual-based bootstrap is proposed, and the first-order consistency of the bootstrap is established. A set of Monte Carlo simulations shows that the bootstrap corrects size distortion of asymptotic distribution in finite samples, and that its power against the threshold cointegration alternative is significantly greater than that of conventional cointegration tests. Our method is illustrated with used car price indexes.     

Limited information Bayesian analysis of a simultaneous equation with an autocorrelated error term and its application to the U.S. gasoline market

Using Markov Chain Monte Carlo algorithms within the limited information Bayesian framework, we estimate the parameters of the structural equation of interest and test weak exogeneity in a simultaneous equation model with white noise as well as autocorrelated error terms. A numerical example and an estimation of the supply and demand equations of the U.S. gasoline market show that if we ignore autocorrelation we obtain unreasonable posterior distributions of the parameters of interest. Also we find that the hypothesis of the asymmetric effect of the changes in oil price on the changes in gasoline price is rejected. Oil inventory has a significant negative effect on the gasoline price.     

A Gaussian approximation scheme for computation of option prices in stochastic volatility models

We consider European options on a price process that follows the log-linear stochastic volatility model. Two stochastic integrals in the option pricing formula are costly to compute. We derive a central limit theorem to approximate them. At parameter settings appropriate to foreign exchange data our formulas improve computation speed by a factor of 1000 over brute force Monte Carlo making MCMC statistical methods practicable. We provide estimates of model parameters from daily data on the Swiss Franc to Euro and Japanese Yen to Euro over the period 1999–2002.     

Common cyclical features analysis in VAR models with cointegration

The paper considers n-dimensional VAR models for variables exhibiting cointegration and common cyclical features. Two specific reduced rank vector error correction models are discussed. In one, named the “strong form” and denoted by SF, the collection of all coefficient matrices of a VECM has rank less than n, in the other, named the “weak form” and denoted by WF, the collection of all coefficient matrices except the matrix of coefficient of error correction terms has rank less than n. The paper explores the theoretical connections between these two forms, suggests asymptotic tests for each form and examines the small sample properties of these tests by Monte Carlo simulations.     

Combining VAR and DSGE forecast densities

A popular macroeconomic forecasting strategy utilizes many models to hedge against instabilities of unknown timing; see (among others) Stock and Watson (2004), Clark and McCracken (2010), and Jore et al. (2010). Existing studies of this forecasting strategy exclude dynamic stochastic general equilibrium (DSGE) models, despite the widespread use of these models by monetary policymakers. In this paper, we use the linear opinion pool to combine inflation forecast densities from many vector autoregressions (VARs) and a policymaking DSGE model. The DSGE receives a substantial weight in the pool (at short horizons) provided the VAR components exclude structural breaks. In this case, the inflation forecast densities exhibit calibration failure. Allowing for structural breaks in the VARs reduces the weight on the DSGE considerably, but produces well-calibrated forecast densities for inflation.     

Likelihood based testing for no fractional cointegration

We consider two likelihood ratio tests, the so-called maximum eigenvalue and trace tests, for the null of no cointegration when fractional cointegration is allowed under the alternative, which is a first step to generalize the so-called Johansen’s procedure to the fractional cointegration case. The standard cointegration analysis only considers the assumption that deviations from equilibrium can be integrated of order zero, which is very restrictive in many cases and may imply an important loss of power in the fractional case. We consider the alternative hypotheses with equilibrium deviations that can be mean reverting with order of integration possibly greater than zero. Moreover, the degree of fractional cointegration is not assumed to be known, and the asymptotic null distribution of both tests is found when considering an interval of possible values. The power of the proposed tests under fractional alternatives and size accuracy provided by the asymptotic distribution in finite samples are investigated.     

Nonparametric Bayesian modelling of monotone preferences for discrete choice experiments

Discrete choice experiments are widely used to learn about the distribution of individual preferences for product attributes. Such experiments are often designed and conducted deliberately for the purpose of designing new products. There is a long-standing literature on nonparametric and Bayesian modelling of preferences for the study of consumer choice when there is a market for each product, but this work does not apply when such markets fail to exist as is the case with most product attributes. This paper takes up the common case in which attributes can be quantified and preferences over these attributes are monotone. It shows that monotonicity is the only shape constraint appropriate for a utility function in these circumstances. The paper models components of utility using a Dirichlet prior distribution and demonstrates that all monotone nondecreasing utility functions are supported by the prior. It develops a Markov chain Monte Carlo algorithm for posterior simulation that is reliable and practical given the number of attributes, choices and sample sizes characteristic of discrete choice experiments. The paper uses the algorithm to demonstrate the flexibility of the model in capturing heterogeneous preferences and applies it to a discrete choice experiment that elicits preferences for different auto insurance policies.     

Inference on impulse response functions in structural VAR models

Skepticism toward traditional identifying assumptions based on exclusion restrictions has led to a surge in the use of structural VAR models in which structural shocks are identified by restricting the sign of the responses of selected macroeconomic aggregates to these shocks. Researchers commonly report the vector of pointwise posterior medians of the impulse responses as a measure of central tendency of the estimated response functions, along with pointwise 68% posterior error bands. It can be shown that this approach cannot be used to characterize the central tendency of the structural impulse response functions. We propose an alternative method of summarizing the evidence from sign-identified VAR models designed to enhance their practical usefulness. Our objective is to characterize the most likely admissible model(s) within the set of structural VAR models that satisfy the sign restrictions. We show how the set of most likely structural response functions can be computed from the posterior mode of the joint distribution of admissible models both in the fully identified and in the partially identified case, and we propose a highest-posterior density credible set that characterizes the joint uncertainty about this set. Our approach can also be used to resolve the long-standing problem of how to conduct joint inference on sets of structural impulse response functions in exactly identified VAR models. We illustrate the differences between our approach and the traditional approach for the analysis of the effects of monetary policy shocks and of the effects of oil demand and oil supply shocks.     

The multi-state latent factor intensity model for credit rating transitions

A new empirical reduced-form model for credit rating transitions is introduced. It is a parametric intensity-based duration model with multiple states and driven by exogenous covariates and latent dynamic factors. The model has a generalized semi-Markov structure designed to accommodate many of the stylized facts of credit rating migrations. Parameter estimation is based on Monte Carlo maximum likelihood methods for which the details are discussed in this paper. A simulation experiment is carried out to show the effectiveness of the estimation procedure. An empirical application is presented for transitions in a 7 grade rating system. The model includes a common dynamic component that can be interpreted as the credit cycle. Asymmetric effects of this cycle across rating grades and additional semi-Markov dynamics are found to be statistically significant. Finally, we investigate whether the common factor model suffices to capture systematic risk in rating transition data by introducing multiple factors in the model.