Bayesian Inference in Structural Second-Price Auctions

The aim of this thesis is to develop efficient and practically useful Bayesian methods for statistical inference in structural second-price auctions. The models are applied to a carefully collected coin auction dataset with bids and auction-specific characteristics from one thousand Internet auctions on eBay. Bidders are assumed to be risk-neutral and symmetric, and compete for a single object using the same game-theoretic strategy. A key contribution in the thesis is the derivation of very accurate approximations of the otherwise intractable equilibrium bid functions under different model assumptions. These easily computed and numerically stable approximations are shown to be crucial for statistical inference, where the inverse bid functions typically needs to be evaluated several million times. In the first paper, the approximate bid is a linear function of a bidder's signal and a Gaussian common value model is estimated. We find that the publicly available book value and the condition of the auctioned object are important determinants of bidders' valuations, while eBay's detailed seller information is essentially ignored by the bidders. In the second paper, the Gaussian model in the first paper is contrasted to a Gamma model that allows intrinsically non-negative common values. The Gaussian model performs slightly better than the Gamma model on the eBay data, which we attribute to an almost normal or at least symmetrical distribution of valuations. The third paper compares the model in the first paper to a directly comparable model for private values. We find many interesting empirical regularities between the models, but no strong and consistent evidence in favor of one model over the other. In the last paper, we consider auctions with both private-value and common-value bidders. The equilibrium bid function is given as the solution to an ordinary differential equation, from which we derive an approximate inverse bid as an explicit function of a given bid. The paper proposes an elaborate model where the probability of being a common value bidder is a function of covariates at the auction level. The model is estimated by a Metropolis-within-Gibbs algorithm and the results point strongly to an active influx of both private-value and common-value bidders.At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Epub ahead of print. Paper 2: Manuscript. Paper 3: Manuscript. Paper 4: Manuscript.</p

Bayesian Inference in Structural Second-Price common Value Auctions

Structural econometric auction models with explicit game-theoretic modeling of bidding strategies have been quite a challenge from a methodological perspective, especially within the common value framework. We develop a Bayesian analysis of the hierarchical Gaussian common value model with stochastic entry introduced by Bajari and Hortaçsu (2003). A key component of our approach is an accurate and easily interpretable analytical approximation of the equilibrium bid function, resulting in a fast and numerically stable evaluation of the likelihood function. The analysis is also extended to situations with positive valuations using a hierarchical Gamma model. We use a Bayesian variable selection algorithm that simultaneously samples the posterior distribution of the model parameters and does inference on the choice of covariates. The methodology is applied to simulated data and to a carefully collected dataset from eBay with bids and covariates from 1000 coin auctions. It is demonstrated that the Bayesian algorithm is very efficient and that the approximation error in the bid function has virtually no effect on the model inference. Both models fit the data well, but the Gaussian model outperforms the Gamma model in an out-of-sample forecasting evaluation of auction prices.Bid function approximation; eBay; Internet auctions; Likelihood inference; Markov Chain Monte Carlo; Normal valuations; Variable selection

Bayesian Rician Regression for Neuroimaging

It is well-known that data from diffusion weighted imaging (DWI) follow the Rician distribution. The Rician distribution is also relevant for functional magnetic resonance imaging (fMRI) data obtained at high temporal or spatial resolution. We propose a general regression model for non-central chi (NC-chi) distributed data, with the heteroscedastic Rician regression model as a prominent special case. The model allows both parameters in the Rician distribution to be linked to explanatory variables, with the relevant variables chosen by Bayesian variable selection. A highly efficient Markov chain Monte Carlo (MCMC) algorithm is proposed to capture full model uncertainty by simulating from the joint posterior distribution of all model parameters and the binary variable selection indicators. Simulated regression data is used to demonstrate that the Rician model is able to detect the signal much more accurately than the traditionally used Gaussian model at low signal-to-noise ratios. Using a diffusion dataset from the Human Connectome Project, it is also shown that the commonly used approximate Gaussian noise model underestimates the mean diffusivity (MD) and the fractional anisotropy (FA) in the single-diffusion tensor model compared to the Rician model.Funding Agencies|Swedish research council [2013-5229, 2015-05356]; Information Technology for European Advancement (ITEA) 3 Project BENEFIT; National Institute of Dental and Craniofacial Research (NIDCR); National Institute of Mental Health (NIMH); National Institute of Neurological Disorders and Stroke (NINDS)See also https://doi.org/10.1101/095844, BioRxvi.org</p

Bayesian modelling of effective and functional brain connectivity using hierarchical vector autoregressions

Analysis of brain connectivity is important for understanding how information is processed by the brain. We propose a novel Bayesian vector autoregression hierarchical model for analysing brain connectivity within resting-state functional magnetic resonance imaging, and apply it to simulated data and a real data set with subjects in different groups. Our approach models functional and effective connectivity simultaneously and allows for both group- and single-subject inference. We combine analytical marginalization with Hamiltonian Monte Carlo to obtain highly efficient posterior sampling. We show that our model gives similar inference for effective connectivity compared to models with a common covariance matrix to all subjects, but more accurate inference for functional connectivity between regions compared to models with more restrictive covariance structures. A Stan implementation of our model is available on GitHub

Bayesian Rician Regression for Neuroimaging

It is well-known that data from diffusion weighted imaging (DWI) follow the Rician distribution. The Rician distribution is also relevant for functional magnetic resonance imaging (fMRI) data obtained at high temporal or spatial resolution. We propose a general regression model for non-central χ (NC-χ) distributed data, with the heteroscedastic Rician regression model as a prominent special case. The model allows both parameters in the Rician distribution to be linked to explanatory variables, with the relevant variables chosen by Bayesian variable selection. A highly efficient Markov chain Monte Carlo (MCMC) algorithm is proposed to capture full model uncertainty by simulating from the joint posterior distribution of all model parameters and the binary variable selection indicators. Simulated regression data is used to demonstrate that the Rician model is able to detect the signal much more accurately than the traditionally used Gaussian model at low signal-to-noise ratios. Using a diffusion dataset from the Human Connectome Project, it is also shown that the commonly used approximate Gaussian noise model underestimates the mean diffusivity (MD) and the fractional anisotropy (FA) in the single-diffusion tensor model compared to the Rician model

Identification of potentially relevant metals for the etiology of autism by using a Bayesian multivariate approach for partially censored values

Heavy metals are known to be able to cross the placental and blood brain barriers to affect critical neurodevelopmental processes in the fetus. We measured metal levels (Al, Cd, Hg, Li, Pb and Zn) in the cord blood of newborns and in the serum of the same children at 5 years of age, and compared between individuals with or without (controls) autism spectrum disorder (ASD) diagnosis. The samples were from a biobank associated with the All Babies in Southeast Sweden (ABIS) registry. We proposed a Bayesian multivariate log-normal model for partially censored values to identify potentially relevant metals for the etiology of ASD. Our results in cord blood suggest prenatal Al levels could be indicative of later ASD incidence, which could also be related to an increased possibility of a high, potentially toxic, exposure to Al and Li during pregnancy. In addition, a larger possibility of a high, potentially beneficial, exposure to Zn could occur during pregnancy in controls. Finally, we found decisive evidence for an average increase of Hg in 5-year-old ASD children compared to only weak evidence for controls. This is concordant with previous research showing an impaired ability for eliminating Hg in the ASD group

Bayesian Inference in Structural Second-Price Common Value Auctions

Structural econometric auction models with explicit game-theoretic modeling of bidding strategies have been quite a challenge from a methodological perspective, especially within the common value framework. We develop a Bayesian analysis of the hierarchical Gaussian common value model with stochastic entry introduced by Bajari and Hortaçsu (2003). A key component of our approach is an accurate and easily interpretable analytical approximation of the equilibrium bid function, resulting in a fast and numerically stable evaluation of the likelihood function. The analysis is also extended to situations with positive valuations using a hierarchical Gamma model. We use a Bayesian variable selection algorithm that simultaneously samples the posterior distribution of the model parameters and does inference on the choice of covariates. The methodology is applied to simulated data and to a carefully collected dataset from eBay with bids and covariates from 1000 coin auctions. It is demonstrated that the Bayesian algorithm is very efficient and that the approximation error in the bid function has virtually no effect on the model inference. Both models fit the data well, but the Gaussian model outperforms the Gamma model in an out-of-sample forecasting evaluation of auction prices