The Applied Statistics Workshop 2021

1. The role of the prior in estimating VAR models with sign restrictions

Several recent studies have expressed concern that the Haar prior typically imposed in estimating sign-identified VAR models may be unintentionally informative about the implied prior for the structural impulse responses. This question is indeed important, but we show that the tools that have been used in the literature to illustrate this potential problem are invalid. Specifically, we show that it does not make sense from a Bayesian point of view to characterize the impulse response prior based on the distribution of the impulse responses conditional on the maximum likelihood estimator of the reduced-form parameters, since the the prior does not, in general, depend on the data. We illustrate that this approach tends to produce highly misleading estimates of the impulse response priors. We formally derive the correct impulse response prior distribution and show that there is no evidence that typical sign-identified VAR models estimated using conventional priors tend to imply unintentionally informative priors for the impulse response vector or that the corresponding posterior is dominated by the prior. Our evidence suggests that concerns about the Haar prior for the rotation matrix have been greatly overstated and that alternative estimation methods are not required in typical applications. Finally, we demonstrate that the alternative Bayesian approach to estimating sign-identified VAR models proposed by Baumeister and Hamilton (2015) suffers from exactly the same conceptual shortcoming as the conventional approach. We illustrate that this alternative approach may imply highly economically implausible impulse response priors.

 2. Joint Bayesian Inference about Impulse Responses in VAR Models

We derive the Bayes estimator of vectors of structural VAR impulse responses under a range of alternative loss functions. We also derive joint credible regions for vectors of impulse responses as the lowest posterior risk region under the same loss functions. We show that conventional impulse response estimators such as the posterior median response function or the posterior mean response function are not in general the Bayes estimator of the impulse response vector obtained by stacking the impulse responses of interest. We show that such pointwise estimators may imply response function shapes that are incompatible with any possible parameterization of the underlying model. Moreover, conventional pointwise quantile error bands are not a valid measure of the estimation uncertainty about the impulse response vector because they ignore the mutual dependence of the responses. In practice, they tend to understate substantially the estimation uncertainty about the impulse response vector.

行列空間上で定義された確率分布にたいするマルコフ連鎖モンテカルロ(MCMC)アルゴリズムを研究する。このような重要なサンプリング問題は、解析的に検討しきれていない。関連するMCMCアルゴリズムのエルゴード性を解析することで、この分野への大きな一歩を踏み出す。標準的なランダムウォーク型メトロポリス法（RWM）とpCN法に加えて、本研究では（行列版の）「混合された（Mixed）」pCN法 = MpCN法と呼ばれる新しいアルゴリズムを提案する。RWMとpCNは、重い裾を持つ行列分布の重要なクラスにたいして、指数エルゴード的でないことが示された。対照的に、MpCNは異なる裾の特性を持つ確率分布にたいして頑健であり、裾の重い分布のクラスにおいて非常に優れた経験的性能を持つ。MpCNの指数エルゴード性は、この研究では完全には証明されていない。ドリフト条件のいくつかは、状態空間の複雑さのために得るのが非常に困難だからだ。 しかし、証明に向けて大きく前進し、今後の課題として残された最後のステップを詳細に示している。 また、金融統計学で生じる困難なモデルの実データでの数値応用を通じて、アルゴリズムの計算性能を説明する。

本研究はAlexandros Beskos (University College London)との共同研究です。

東芝との共同研究において，製造業の課題を動機として研究が進み，論文化にまで至った研究を2つ紹介したい． 1つは高次元高欠測データの解析である．90%以上もの欠測が生じる特徴量がある．そのような場合に妥当に回帰モデルを推定する問題を考えた．CoCoLassoという手法はその問題に対応できるのだが，高欠測の場合をあまり意識していないと考えた．CoCOLassoに欠測率に応じた重み補正を行うことで，より妥当な手法を提案した．最適な重みについては理論的な考察も行った． もう1つは転移学習に関わる研究である．Lassoを繰り返し使う状況において，以前の情報を有効活用したい，パラメータ推定値の変化が小さくなるようにしたい，現場の経験を活かしたい，という要望を満たす手法を開発した．理論解析も行って手法の良さを理論的にサポートした．最後に，それらの成果がメディア等にどのように取り上げられたかについても触れたい． 

Sparse signal recovery remains an important challenge in large scale data analysis and global-local (G-L) shrinkage priors have undergone an explosive development in the last decade in both theory and methodology. These developments have established the G-L priors as the state-of-the-art Bayesian tool for sparse signal recovery as well as default non-linear problems. In the first half of my talk, I will survey the recent advances in this area, focusing on optimality and performance of G-L priors for both continuous as well as discrete data. In the second half, I will discuss several recent developments, including designing a shrinkage prior to handle bi-level sparsity in regression and handling sparse compositional data, routinely observed in microbiomics. I will discuss the methodological challenges associated with each of these problems, and propose to address this gap by using new prior distributions, specially designed to enable handling structured data. I will provide some theoretical support for the proposed methods and show improved performance in simulation settings and application to environmentrics and microbiome data.

In this paper, we propose a class of multivariate non-Gaussian time series models which include dynamic versions of many well-known distributions and consider their Bayesian analysis. A key feature of our proposed model is its ability to account for correlations across time as well as across series (contemporary) via a common random environment. The proposed modeling approach yields analytically tractable dynamic marginal likelihoods, a property not typically found outside of linear Gaussian time series models. These dynamic marginal likelihoods can be tied back to known static multivariate distributions such as the Lomax, generalized Lomax, and the multivariate Burr distributions. The availability of the marginal likelihoods allows us to develop efficient estimation methods for various settings using Markov chain Monte Carlo as well as sequential Monte Carlo methods. Our approach can be considered to be a multivariate generalization of commonly used univariate non-Gaussian class of state space models. To illustrate our methodology, we use simulated data examples and a real application of multivariate time series for modeling the joint dynamics of stochastic volatility in financial indexes, the VIX and VXN.