Bayesian econometrics pdf

By employing the … Expand. View 2 excerpts, references background. An Introduction to Modern Bayesian Econometrics. The Bayesian Algorithm. Prediction and Model Checking. Linear Regression. Bayesian Calculations. Nonlinear Regression Models. Randomized, Controlled and Observational … Expand.

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With proper adjustments, our proposed idea could be applied to GARCH models and to all related models for which it is easier to sample from the non-identifiable version of the model. Approaches based on DPM have alreay been used in the econometric literature, for instance by Jensen and Maheu to model volatility distribution, or by Kalli et al. Along the same lines, Jensen and Maheu and Jensen and Maheu propose a DPM approach to stochastic volatility and financial returns, and Zaharieva et al.

The rest of the paper is organized as follows. The model is evaluated in terms of fitting and predictive performance against parametric alternatives in a simulation study Section 4 and in an empirical financial application on the interaction among volatility measures Section 5.

In the Section 6 we conclude and propose further developments. The unit mean assumption on the innovation term second bullet above is necessary to guarantee the identifiability of the model. The i. Yet this is not a necessary condition and, in multivariate context, it is also quite restrictive.

Several generalizations of this specification has been proposed. To study inter-dependencies across volatility measures, Cipollini et al. Engle et al. There are two important problems with finite component mixtures: it is usually difficult to determine a priori the required number of components and they lack the degree of flexibility that is needed in many applications.

Therefore this approach bypasses the problem of choosing the correct number of components. Furthermore, Dalal and Hall establish the large support property, or adequacy, of DPM, in that a parametric Bayesian model can be approximated by a nonparametric Bayesian model with a mixture of DPs, with the prior assigning most of its weight to neighborhoods of the parametric model, and show that any parametric or nonparametric prior may be approximated arbitrarily closely by a prior which is a mixture of DPs.

See also Peluso et al. Corollary 2. As pointed out by Kalli et al. An alternative could be to consider more general stick-breaking processes, for more details refer to Kalli et al.

A multivariate log-normal distribution is preferred to a multivariate Gamma, that would had been the direct multivariate generalization of the univariate approach suggested by Solgi and Mira , since all multivariate generalizations of the Gamma distribution we are aware of are defined via the joint characteristic function and thus require numerical inversion formulas to find the corresponding density.

As mentioned earlier, in parametric vMEM the distribution of innovations is restricted to have unit-vector mean. Imposing all component means to only depend on the diagonal elements of the component- specific scale matrix restricts in some way the ability of the model to cover all the possible distri- butions on the positive orthant.

In fact, in the univariate log-normal case and in the univariate Gamma case with one parameter of Solgi and Mira the introduction of components with thicker right tails will increase the probability of the neighbourhood around zero, to keep the mean fixed. Hence, in presence of fat-tailed innovation errors, while this univariate DPM attempts to assign higher weights to the components with smaller precision, it will, at the same time, increases the likelihood of innovations close to zero.

In the multivariate case, the same reasoning is valid for marginals and extended to the joint distribution. As a consequence, this model does not effectively range over all the possibly true distributions on the positive orthant. Therefore the mean vector decays exponentially with j and thus the sum should be finite.

Following this idea, for the purpose of the sampling algorithm, we start from the unconstrained DPM for the distribution of the innovations, which is a parameter expanded in the sense of Liu J. It is important to enlighten that a prior on the parameters of the PX model induces a prior on the parameters of the original model and that the use of proper priors results in proper posteriors for this model even if the likelihood is improper.

Here we will describe, adapt to our goals and finally use its efficient version due to Kalli et al. In order to improve the efficiency of the slice sampler, Kalli et al. Kalli et al. In their examples, Kalli et al. We will now detail the steps of the slice sampler. Thus the full conditional probability density function of vj is Y p vj. Otherwise the full conditional of vj is equal to the prior distribution. Therefore at this step of the sampling we only need to update a finite number, N , of vj s: the others will not be updated and, if we will ever need them in other steps of the sampler, we will sample them from their prior.

The simulation time on a server running at 2. We finally repeat the whole procedure for 40 datasets. All the estimates are based on effective sample sizes greater than In Figures 1, 2, 3 we reported the traces, the posterior densities and the autocorrelation functions of the post-processed parameters of the conditional mean for a randomly chosen dataset.

These figures show that all the traces have reached convergence and all the autocorrelation functions become non-significant in less than lags, and most of them in less than We can see that there are on average 6 active components but, correctly, only two of them have really significant weights.

The repetitions over the 40 different datasets, whose figures are not reported for brevity, provide the same results: BSP-vMEM performs better than the simple parametric model in approximating the pdf of the innovations and the OSA pdf, and same convergence behaviour for the model pa- rameters. Same considerations hold when we compare the proposed method with the Maximum a Posteriori of the Bayesian model with no DPM on the innovation error: the Log Pseudo Marginal Likelihood, estimated as suggested in Nieto-Barajas et al.

True Est. Figure 4: The upper left plot shows the traces of total number of components and of the number of active components at each step.

The lower left plot shows the corresponding running averages. The plot on the right shows the traces of the mixture weights. Since then the literature has significantly expanded: from the realized volatility of Andersen and Bollerslev and Peluso et al. In parallel to the evolution of these measures, there has been a natural complementary effort to build adequate models to describe their dynamics. Multiplicative Error Models have been used for this purpose for example by Cipollini et al.

The time series which are most commonly used in this respect are the squared close-to-close adjusted returns rt2 , the realized variances rvt2 in any of their flavours , the absolute returns rt , the realized volatilities rvt , and the daily ranges hlt. For our analysis we make use of a bivariate series composed by daily absolute returns and realized kernel volatilities, rt , rvt.

From the time series plotted in Figure 6, we can see that both the measures of all the three indices share some common features like alternance of periods of high and low volatility and persistence. For all the time series, we run , iterations of the algorithm described in Section 3 and then discard the first 30, of them as burn-in.

In all the analyses, all the effective sample sizes of the variables obtained from the MCMC simulations are bigger than Very similar figures have been obtained for other parameters and for the other two time series. MAP Est. MCMC Est. Figure 8: Estimated joint and marginal densities of the innovations over the estimated innovations. The lower plot shows the corresponding running averages. From Figure 7 it can be seen that, although there is some autocorrelation the traces of the MCMC simulations have all reached convergence and the posterior histograms look informative.

For all the three time series and both the models considered, we obtain some common qualitative features of the point estimates. Second, the estimates of the coefficients of the second column of matrix A are always bigger, in absolute value, than the ones in the first column of the same matrix. This suggests that the lagged realizations of the realized kernel volatility influence the evolution of both the components of the conditional mean vector more than the lagged realizations of the absolute returns.

This fact, that could look strange at first sight, simply means that the lagged observation of the realized volatility contains more information on the present realization of the conditional mean of the absolute returns than the lagged absolute returns and this can be viewed as a further proof of the fact that the realized volatilities are more informative about the latent volatility than the absolute returns.

In all the empirical analyses we tried there have always been about the same average number of active and total components of the DPM, for all the time series. The sum of the average of the weights of these components is always bigger than 0. Our contribution is the formulation of a statistical model that is i not bounded to special parametric forms of the error term distribution, known to be a quite strong restriction with multivariate data, and ii subject to weaker assumptions than the other semiparametric approaches in the literature, based on the Generalized Method of Moments.

In more details, the innovation term of our vMEM follows a location-scale DPM of multivariate log-normal distributions. By exploiting a parameter-expanded unconstrained version of the model, we are able to simplify the computational difficulties arising from the constraints to the positive or- thant and we formalize an efficient slice sampler for posterior inference. The proposed model shows better fitting and predictive performances than its parametric counterpart in both the simulations and in the empirical study on the interaction between daily absolute returns and realized kernel volatilities.

Further developments of interest include i a refinement of the sampling technique with sparsity- driven efficiencies that can manage time series in high dimension, ii a more complex specification of the conditional mean, with the purpose of comparing volatility proxies through more elaborated models with non-linearities in the dynamics and iii the adoption of the proposed model for other applications of interest, e.

Acknowledgements The authors gratefully thank Reza Solgi for sharing the details of his previous research on univariate multiplicative error morels, Maria Conception Ausin for fruitful discussions on the sampling aspects of the proposed algorithm, Fabrizio Leisen, Antonio Canale, Bernardo Nipoti, Jim Griffin and Sonia Petrone for useful comments on intermediate results.

High-frequency covariance estimates with noisy and asynchronous financial data. The concept of risk 3. Overview of count response models 4.

Methods of estimation and assessment 5. Assessment of count models 6. Poisson regression 7. Overdispersion 8. Robust bayesian model Selection. It is shown that when a model is … Expand. In this supplement, we provide additional posterior results that complement those documented in Section 4 of the main text. Specically, we report Bayesian point and interval estimates of the static … Expand.

View 1 excerpt, cites methods. Consensus priors for multinomial and binomial ratios. Reference analysis is an objective Bayesian approach to finding non-informative prior distributions.

For various models involving nuisance parameters the reference prior has been shown to be superior … Expand. Bayesian inference in dynamic discrete choice models. Highly Influenced. View 3 excerpts, cites background and methods. Covariance, identification, and finite-sample performance of the MSL and Bayes estimators of a logit model with latent attributes.

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Estimation of unknown parameters and functions involved in complex nonlinear econometric models is a very important issue. Existing estimation methods include generalised method of moments GMM by … Expand. View 2 excerpts, cites methods and background. Bayesian modeling of joint and conditional distributions.

In this paper, we study a Bayesian approach to exible modeling of conditional distributions.