MC's for MCMC'ists

We develop a minimum amount of theory of Markov chains at as low a level of abstraction as possible in order to prove two fundamental probability laws for standard Markov chain Monte Carlo algorithms:
1. The law of large numbers explains why the algorithm works: it states that the empirical means calculated from the samples converge towards their "true" expected values, viz. expectations with respect to the invariant distribution of the associated Markov chain (=the target distribution of the simulation).
2. The central limit theorem expresses the deviations of the empirical means from their expected values in terms of asymptotically normally distributed random variables. We also present a formula and an estimator for the associated variance.     

Bayesian time series analysis of periodic behaviour and spectral structure

Analysis, model selection and forecasting in univariate time series models can be routinely carried out for models in which the model order is relatively small. Under an ARMA assumption, classical estimation, model selection and forecasting can be routinely implemented with the Box–Jenkins time domain representation. However, this approach becomes at best prohibitive and at worst impossible when the model order is high. In particular, the standard assumption of stationarity imposes constraints on the parameter space that are increasingly complex. One solution within the pure AR domain is the latent root factorization in which the characteristic polynomial of the AR model is factorized in the complex domain, and where inference questions of interest and their solution are expressed in terms of the implied (reciprocal) complex roots; by allowing for unit roots, this factorization can identify any sustained periodic components. In this paper, as an alternative to identifying periodic behaviour, we concentrate on frequency domain inference and parameterize the spectrum in terms of the reciprocal roots, and, in addition, incorporate Gegenbauer components. We discuss a Bayesian solution to the various inference problems associated with model selection involving a Markov chain Monte Carlo (MCMC) analysis. One key development presented is a new approach to forecasting that utilizes a Metropolis step to obtain predictions in the time domain even though inference is being carried out in the frequency domain. This approach provides a more complete Bayesian solution to forecasting for ARMA models than the traditional approach that truncates the infinite AR representation, and extends naturally to Gegenbauer ARMA and fractionally differenced models.     

Evolving densities in continuous strategy games through particle simulations

Many cases of strategic interaction between agents involve a continuous set of choices. It is natural to model these problems as continuous space games. Consequently, the population of agents playing the game will be represented with a density function defined over the continuous set of strategy choices. Simulating evolutionary dynamics on continuous strategy spaces is a challenging problem. The classic approach of discretizing the strategy space is ineffective for multidimensional strategy spaces. We present a principled approach to simulation of adaptive dynamics in continuous space games using sequential Monte Carlo methods. Sequential Monte Carlo methods use a set of weighted random samples, also named particles to represent density functions over multidimensional spaces. Sequential Monte Carlo methods provide computationally efficient ways of computing the evolution of probability density functions. We employ resampling and smoothing steps to prevent particle degeneration problem associated with particle estimates. The resulting algorithm can be interpreted as an agent based simulation with elements of natural selection, regression to mean and mutation. We illustrate the performance of the proposed simulation technique using two examples: continuous version of the repeated prisoner dilemma game and evolution of bidding functions in first-price closed-bid auctions.     

The Effect of Market Regimes on Style Allocation

We analyse time-varying risk premia and the implications for portfolio choice. Using Markov Chain Monte Carlo (MCMC) methods, we estimate a multivariate regime-switching model for the Carhart (1997) four-factor model. We find two clearly separable regimes with different mean returns, volatilities, and correlations. In the High-Variance Regime, only value stocks deliver a good performance, whereas in the Low-Variance Regime, the market portfolio and momentum stocks promise high returns. Regime-switching induces investors to change their portfolio style over time depending on the investment horizon, the risk aversion, and the prevailing regime. Value investing seems to be a rational strategy in the High-Variance Regime, momentum investing in the Low-Variance Regime. An empirical out-of-sample backtest indicates that this switching strategy can be profitable, but the overall forecasting ability for the regime-switching model seems to be weak compared to the iid model.     

Forecasting and turning point predictions in a Bayesian panel VAR model

We provide methods for forecasting variables and predicting turning points in panel Bayesian VARs. We specify a flexible model, which accounts for both interdependencies in the cross section and time variations in the parameters. Posterior distributions for the parameters are obtained for hierarchical and for Minnesota-type priors. Formulas for multistep, multiunit point and average forecasts are provided. An application to the problem of forecasting the growth rate of output and of predicting turning points in the G-7 illustrates the approach. A comparison with alternative forecasting methods is also provided.     

Performance of empirical Bayes estimators of random coefficients in multilevel analysis: Some results for the random intercept-only model

For a multilevel model with two levels and only a random intercept, the quality of different estimators of the random intercept is examined. Analytical results are given for the marginal model interpretation where negative estimates of the variance components are allowed for. Except for four or five level-2 units, the Empirical Bayes Estimator (EBE) has a lower average Bayes risk than the Ordinary Least Squares Estimator (OLSE). The EBEs based on restricted maximum likelihood (REML) estimators of the variance components have a lower Bayes risk than the EBEs based on maximum likelihood (ML) estimators. For the hierarchical model interpretation, where estimates of the variance components are restricted being positive, Monte Carlo simulations were done. In this case the EBE has a lower average Bayes risk than the OLSE, also for four or five level-2 units. For large numbers of level-1 (30) or level-2 units (100), the performances of REML-based and ML-based EBEs are comparable. For small numbers of level-1 (10) and level-2 units (25), the REML-based EBEs have a lower Bayes risk than ML-based EBEs only for high intraclass correlations (0.5).     

Bayesian MCMC Mapping of Quantitative Trait Loci in a Half-sib Design: a Graphical Model Perspective

Graphical models provide a powerful and flexible approach to the analysis of complex problems in genetics. While task-specific software may be extremely efficient for any particular analysis, it is often difficult to adapt to new computational challenges. By viewing these genetic applications in a more general framework, many problems can be handled by essentially the same software. This is advantageous in an area where fast methodological development is essential. Once a method has been fully developed and tested, problem-specific software may then be required. The aim of this paper is to illustrate the potential use of a graphical model approach to genetic analyses by taking a very simple and well-understood problem by way of example.     

A comparison of several approximations to the power function of the likelihood ratio test in covariance structure analysis*

In this paper we compare alternative asymptotic approximations to the power of the likelihood ratio test used in covariance structure analysis for testing the fit of a model. Alternative expressions for the noncentrality parameter (ncp) lead to different approximations to the power function. It appears that for alternative covariance matrices close to the null hypothesis, the alternative ncp's lead to similar values, while for alternative covariance matrices far from H_o the different expressions for the ncp can conflict substantively. Monte Carlo evidence shows that the ncp proposed in Satorra and Saris (1985) gives the most accurate power approximations.     

基于蒙特卡罗模拟的商业银行信用风险度量方法

通过与Matlab程序相结合的方式介绍了基于蒙特卡罗模拟的商业银行信用风险度量方法。该方法使在给定的置信水平下科学地估算国内商业银行的信用风险成为可能。     

Pooling data for the analysis of dynamic marketing systems

Vector autoregressive (VAR) models have become popular in marketing literature for analyzing the behavior of competitive marketing systems. One drawback of these models is that the number of parameters can become very large, potentially leading to estimation problems. Pooling data for multiple cross-sectional units (stores) can partly alleviate these problems. An important issue in such models is how heterogeneity among cross-sectional units is accounted for. We investigate the performance of several pooling approaches that accommodate different levels of cross-sectional heterogeneity in a simulation study and in an empirical application. Our results show that the random coefficients modeling approach is an overall good choice when the estimated VAR model is used for out-of-sample forecasting only. When the estimated model is used to compute Impulse Response Functions, we conclude that one should select a modeling approach that matches the level of heterogeneity in the data.