Forecasting levels of log variables in vector autoregressions

Sometimes forecasts of the original variable are of interest, even though a variable appears in logarithms (logs) in a system of time series. In that case, converting the forecast for the log of the variable to a naïve forecast of the original variable by simply applying the exponential transformation is not theoretically optimal. A simple expression for the optimal forecast under normality assumptions is derived. However, despite its theoretical advantages, the optimal forecast is shown to be inferior to the naïve forecast if specification and estimation uncertainty are taken into account. Hence, in practice, using the exponential of the log forecast is preferable to using the optimal forecast.     

Bayesian inference and prediction in the single server Markovian queue

The use of stochastic models and performance measures for the analysis of real life queuing scenarios are based on the fundamental premise that parameters values are known. This is a rarity since more often than not, parameters are usually unknown and require to be estimated. This paper presents techniques for the same from Bayesian perspective. The queue we intend to deal with is the M/M/1 queuing model. Several closed form expressions on posterior inference and prediction are presented which can be readily implemented using standard spreadsheet tools. Previous work in this direction resulted in non-existence of posterior moments. A way out is suggested. Interval estimates and tests of hypothesis on performance measures are also presented.     

Forecasting using variational Bayesian inference in large vector autoregressions with hierarchical shrinkage

Many recent papers in macroeconomics have used large vector autoregressions (VARs) involving 100 or more dependent variables. With so many parameters to estimate, Bayesian prior shrinkage is vital to achieve reasonable results. Computational concerns currently limit the range of priors used and render difficult the addition of empirically important features such as stochastic volatility to the large VAR. In this paper, we develop variational Bayesian methods for large VARs that overcome the computational hurdle and allow for Bayesian inference in large VARs with a range of hierarchical shrinkage priors and with time-varying volatilities. We demonstrate the computational feasibility and good forecast performance of our methods in an empirical application involving a large quarterly US macroeconomic data set.     

A comparison of two model averaging techniques with an application to growth empirics

Parameter estimation under model uncertainty is a difficult and fundamental issue in econometrics. This paper compares the performance of various model averaging techniques. In particular, it contrasts Bayesian model averaging (BMA) — currently one of the standard methods used in growth empirics — with a new method called weighted-average least squares (WALS). The new method has two major advantages over BMA: its computational burden is trivial and it is based on a transparent definition of prior ignorance. The theory is applied to and sheds new light on growth empirics where a high degree of model uncertainty is typically present.     

A new structural break model,with an application to Canadian inflation forecasting

This paper develops an efficient approach to modelling and forecasting time series data with an unknown number of change-points. Using a conjugate prior and conditioning on time-invariant parameters, the predictive density and the posterior distribution of the change-points have closed forms. Furthermore, the conjugate prior is modeled as hierarchical in order to exploit the information across regimes. This framework allows breaks in the variance, the regression coefficients, or both. The regime duration can be modelled as a Poisson distribution. A new, efficient Markov chain Monte Carlo sampler draws the parameters from the posterior distribution as one block. An application to a Canadian inflation series shows the gains in forecasting precision that our model provides.     

Bayesian median autoregression for robust time series forecasting

We develop a Bayesian median autoregressive (BayesMAR) model for time series forecasting. The proposed method utilizes time-varying quantile regression at the median, favorably inheriting the robustness of median regression in contrast to the widely used mean-based methods. Motivated by a working Laplace likelihood approach in Bayesian quantile regression, BayesMAR adopts a parametric model bearing the same structure as autoregressive models by altering the Gaussian error to Laplace, leading to a simple, robust, and interpretable modeling strategy for time series forecasting. We estimate model parameters by Markov chain Monte Carlo. Bayesian model averaging is used to account for model uncertainty, including the uncertainty in the autoregressive order, in addition to a Bayesian model selection approach. The proposed methods are illustrated using simulations and real data applications. An application to U.S. macroeconomic data forecasting shows that BayesMAR leads to favorable and often superior predictive performance compared to the selected mean-based alternatives under various loss functions that encompass both point and probabilistic forecasts. The proposed methods are generic and can be used to complement a rich class of methods that build on autoregressive models.     

Empirical prediction intervals revisited

Empirical prediction intervals are constructed based on the distribution of previous out-of-sample forecast errors. Given historical data, a sample of such forecast errors is generated by successively applying a chosen point forecasting model to a sequence of fixed windows of past observations and recording the associated deviations of the model predictions from the actual observations out-of-sample. The suitable quantiles of the distribution of these forecast errors are then used along with the point forecast made by the selected model to construct an empirical prediction interval. This paper re-examines the properties of the empirical prediction interval. Specifically, we provide conditions for its asymptotic validity, evaluate its small sample performance and discuss its limitations.     

Evaluating predictive densities of US output growth and inflation in a large macroeconomic data set

We evaluate conditional predictive densities for US output growth and inflation using a number of commonly-used forecasting models that rely on large numbers of macroeconomic predictors. More specifically, we evaluate how well conditional predictive densities based on the commonly-used normality assumption fit actual realizations out-of-sample. Our focus on predictive densities acknowledges the possibility that, although some predictors can cause point forecasts to either improve or deteriorate, they might have the opposite effect on higher moments. We find that normality is rejected for most models in some dimension according to at least one of the tests we use. Interestingly, however, combinations of predictive densities appear to be approximated correctly by a normal density: the simple, equal average when predicting output growth, and the Bayesian model average when predicting inflation.     

非寿险精算中的贝叶斯信用模型分析

针对传统B櫣hlmann-Straub信用模型不能有效地解决缺失数据信息处理问题,本文利用贝叶斯统计方法,构造了一类新的贝叶斯信用分析模型,引入基于吉布斯抽样的马尔科夫链蒙特卡洛方法进行数值计算,建立了一个索赔后验分层正态模型进行实证分析,证明模型的有效性。研究结果表明,基于MCMC的贝叶斯信用模型能够动态模拟模型参数的后验分布,提高模型估计的精度,对保险公司经验费率厘定方法的改进具有重要的现实意义。     

On model selection in Bayesian regression

We discuss the problem of constructing a suitable regression model from a nonparametric Bayesian viewpoint. For this purpose, we consider the case when the error terms have symmetric and unimodal densities. By the Khintchine and Shepp theorem, the density of response variable can be written as a scale mixture of uniform densities. The mixing distribution is assumed to have a Dirichlet process prior. We further consider appropriate prior distributions for other parameters as the components of the predictive device. Among the possible submodels, we select the one which has the highest posterior probability. An example is given to illustrate the approach.     

On a bayesian criterion for choosing predictive sub-models in linear regression

Summary We treat the model selection problem in regression as a decision problem in which the decisions are the alternative predictive distributions based on the different sub-models and the parameter space is the set of possible future values of the regressand. The loss function balances out the conflicting needs for a predictive distribution with mean close to the true value ofy but without too great a variation. The treatment is Bayesian and the criterion derived is a Bayesian generalization of Mallows (1973)C _p , the Bivar criterion (Young 1982) and AIC (Akaike 1974). An application using a graphical sensitivity analysis is presented.     

Returns to Schooling and Bayesian Model Averaging: A Union of Two Literatures

Abstract.  In this paper, we review and unite the literatures on returns to schooling and Bayesian model averaging. We observe that most studies seeking to estimate the returns to education have done so using particular (and often different across researchers) model specifications. Given this, we review Bayesian methods which formally account for uncertainty in the specification of the model itself, and apply these techniques to estimate the economic return to a college education. The approach described in this paper enables us to determine those model specifications which are most favored by the given data, and also enables us to use the predictions obtained from all of the competing regression models to estimate the returns to schooling. The reported precision of such estimates also account for the uncertainty inherent in the model specification. Using U.S. data from the National Longitudinal Survey of Youth (NLSY), we also revisit several 'stylized facts' in the returns to education literature and examine if they continue to hold after formally accounting for model uncertainty.