Bayesian Inference in the Social Sciences
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List of Figures iii
1 Bayesian Analysis of Dynamic Network Regression with Joint Edge/Vertex Dynamics 1
Zack W. Almquist and Carter T. Butts
1.1 Introduction 2
1.2 Statistical Models for Social Network Data 2
1.3 Dynamic Network Logistic Regression with Vertex Dynamics 11
1.4 Empirical Examples and Simulation Analysis 14
1.5 Discussion 29
1.6 Conclusion 30
2 Ethnic Minority Rule and Civil War: A Bayesian Dynamic Multilevel Analysis 39
Xun Pang
2.1 Introduction: Ethnic Minority Rule and Civil War 40
2.2 EMR: Grievance and Opportunities of Rebellion 41
2.3 Bayesian GLMM-AR(p) Model 42
2.4 Variables, Model and Data 47
2.5 Empirical Results and Interpretation 49
2.6 Civil War: Prediction 54
2.7 Robustness Checking: Alternative Measures of EMR 59
2.8 Conclusion 60
References 62
3 Bayesian Analysis of Treatment Effect Models 67
Mingliang Li and Justin L. Tobias
3.1 Introduction 68
3.2 Linear Treatment Response Models Under Normality 69
3.3 Nonlinear Treatment Response Models 73
3.4 Other Issues and Extensions: Non-Normality, Model Selection and Instrument Imperfection 78
3.5 Illustrative Application 84
3.6 Conclusion 89
4 Bayesian Analysis of Sample Selection Models 95
Martijn van Hasselt
4.1 Introduction 95
4.2 Univariate Selection Models 97
4.3 Multivariate Selection Models 101
4.4 Semiparametric Models 111
4.5 Conclusion 114
References 114
5 Modern Bayesian Factor Analysis 117
Hedibert Freitas Lopes
5.1 Introduction 117
5.2 Normal linear factor analysis 119
5.3 Factor stochastic volatility 125
5.4 Spatial factor analysis 128
5.5 Additional developments 133
5.6 Modern non-Bayesian factor analysis 136
5.7 Final remarks 137
6 Estimation of stochastic volatility models with heavy tails and serial dependence 159
Joshua C.C. Chan and Cody Y.L. Hsiao
6.1 Introduction 159
6.2 Stochastic Volatility Model 160
6.3 Moving Average Stochastic Volatility Model 168
6.4 Stochastic Volatility Models with Heavy-Tailed Error Distributions 173
References 178
7 From the Great Depression to the Great Recession: A Modelbased Ranking of U.S. Recessions 181
Rui Liu and Ivan Jeliazkov
7.1 Introduction 181
7.2 Methodology 183
7.3 Results 188
7.4 Conclusions 191
Appendix: Data 192
References 192
8 What Difference Fat Tails Make: A Bayesian MCMC Estimation of Empirical Asset Pricing Models 201
Paskalis Glabadanidis
8.1 Introduction 202
8.2 Methodology 204
8.3 Data 205
8.4 Empirical Results 206
8.5 Concluding Remarks 212
9 Stochastic Search For Price Insensitive Consumers 227
Eric Eisenstat
9.1 Introduction 228
9.2 Random utility models in marketing applications 230
9.3 The censored mixing distribution in detail 234
9.4 Reference price models with price thresholds 240
9.5 Conclusion 244
References 245
10 Hierarchical Modeling of Choice Concentration of US Households 249
Karsten T. Hansen, Romana Khan and Vishal Singh
10.1 Introduction 250
10.2 Data Description 252
10.3 Measures of Choice Concentration 252
10.4 Methodology 254
10.5 Results 256
10.6 Interpreting ¿ 260
10.7 Decomposing the effects of time, number of decisions and concentration preference 263
10.8 Conclusion 265
References 267
11 Approximate Bayesian inference in models defined through estimating equations 269
11.1 Introduction 269
11.2 Examples 271
11.3 Frequentist estimation 273
11.4 Bayesian estimation 276
11.5 Simulating from the posteriors 281
11.6 Asymptotic theory 283
11.7 Bayesian validity 285
11.8 Application 286
11.9 Conclusions 288
12 Reacting to Surprising Seemingly Inappropriate Results 295
Dale J. Poirier
12.1 Introduction 295
12.2 Statistical Framework 296
12.3 Empirical Illustration 300
12.4 Discussion 301
References 301
13 Identification and MCMC estimation of bivariate probit models with partial observability 303
Ashish Rajbhandari
13.1 Introduction 303
13.2 Bivariate Probit Model 305
13.3 Identification in a partially observable model 307
13.4 Monte Carlo Simulations 308
13.5 Bayesian Methodology 309
13.6 Application 312
13.7 Conclusion 315
Chapter Appendix 316
References 317
14 School Choice Effects in Tokyo Metropolitan Area: A Bayesian Spatial Quantile Regression Approach 321
Kazuhiko Kakamu and Hajime Wago
14.1 Introduction 321
14.2 The Model 323
14.3 Posterior Analysis 325
14.4 Empirical Analysis 326
14.5 Conclusions 330
CHAPTER 1
BAYESIAN ANALYSIS OF DYNAMIC NETWORK REGRESSION WITH JOINT EDGE/VERTEX DYNAMICS
ZACK W. ALMQUIST1 AND CARTER T. BUTTS2
1University of Minnesota, USA.
2University of California, Irvine, USA.
1.1 Introduction
Change in network structure and composition has been a topic of extensive theoretical and methodological interest over the last two decades; however, the effects of endogenous group change on interaction dynamics within the context of social networks is a surprisingly understudied area. Network dynamics may be viewed as a process of change in the edge structure of a network, in the vertex set on which edges are defined, or in both simultaneously. Recently, Almquist and Butts (2014) introduced a simple family of models for network panel data with vertex dynamics-referred to here as dynamic network logistic regression (DNR)-expanding on a subfamily of temporal exponential-family random graph models (TERGM) (see Robins and Pattison, 2001; Hanneke et al., 2010). Here, we further elaborate this existing approach by exploring Bayesian methods for parameter estimation and model assessment. We propose and implement techniques for Bayesian inference via both maximum a posteriori probability (MAP) and Markov chain Monte Carlo (MCMC) under several different priors, with an emphasis on minimally informative priors that can be employed in a range of empirical settings. These different approaches are compared in terms of model fit and predictive model assessment using several reference data sets.
This chapter is laid out as follows: (1) We introduce the standard (exponential family) framework for modeling static social network data, including both MLE and Bayesian estimation methodology; (2) we introduce network panel data models, discussing both MLE and Bayesian estimation procedures; (3) we introduce a subfamily of the more general panel data models (dynamic network logistic regression)-which allows for vertex dynamics-and expand standard MLE procedures to include Bayesian estimation; (4) through simulation and empirical examples we explore the effect of different prior specifications on both parameter estimation/hypothesis tests and predictive adequacy; (5) finally, we conclude with a summary and discussion of our findings.
1.2 Statistical Models for Social Network Data
The literature on statistical models for network analysis has grown substantially over the last two decades (for a brief review see Butts, 2008b). Further, the literature on dynamic networks has expanded extensively in this last decade - a good overview can be found in Almquist and Butts (2014). In this chapter we use a combination of commonly used statistical and graph theoretic notation. First, we briefly introduce necessary notation and literature for the current state of the art in network panel data models, then we review these panel data models in their general form, including their Bayesian representation. Last, we discuss a specific model family (DNR) which reduces to an easily employed regression-like structure, and formalize it to the Bayesian context.
1.2.1 Network Data and Nomenclature
For purposes of this chapter, we will focus on networks (social or otherwise) that can be represented in terms of dichotomous (i.e., unvalued) ties among pairs of discrete entities. [For more general discussion of network representation, see, e.g., Wasserman and Faust (1994); Butts (2009).] We represent the set of potentially interacting entities via a vertex set (V), with the set of interacting pairs (or ordered pairs, for directed relationships) represented by an edge set (E). In combination, these two sets are referred to as a graph, G = (V, E). (Here, we will use the term "graph" generically to refer to either directed or undirected structures, except as indicated otherwise.) Networks may be static, e.g., representing relationships at a single time point or aggregated over a period of time, or dynamic, e.g., representing relationships appearing and disappearing in continuous time or relationship status at particular discrete intervals.
For many purposes, it is useful to represent a graph in terms of its adjacency matrix: for a graph G of order N = |V|, the adjacency matrix Y {0, 1}N × N is a matrix of indicator variables such that Yij = 1 iff the ith vertex of G is adjacent (i.e., sends a tie to) the jth vertex of G. Following convention in the social network (but not graph theoretic) literature, we will refer to N as the size of G.
The above extends naturally to the case of dynamic networks in discrete time. Let us consider the time series ., Gt-1, Gt, Gt+1,., where Gt = (Vt, Et) represents the state of a system of interest at time t. This corresponds in turn to the adjacency matrix series ., Y..t-1, Y..t, Y..t+1,., with Nt = |Vt| being the size of the network at time t and Y..t {0, 1}NtxNt such that Yijt = 1 iff the ith vertex of Gt is adjacent to the jth vertex of Gt at time t. As this notation implies, the vertex set of an evolving network is not necessarily fixed; we shall be particularly interested here in the case in which Vt is drawn from some larger risk set, such that vertices may enter and leave the network over time.
1.2.2 Exponential Family Random Graph Models
When modeling social or other networks, it is often helpful to represent their distributions via random graphs in discrete exponential family form. Graph distributions expressed in this way are called exponential family random graph models or ERGMs. Holland and Leinhardt (1981) are generally credited with the first explicit use of statistical exponential families to represent random graph models for social networks, with important extensions by Frank and Strauss (1986) and subsequent elaboration by Wasserman and Pattison (1996), Pattison and Wasserman (1999), Pattison and Robins (2002), Snijders et al. (2006), Butts (2007), and others. The power of this framework lies in the extensive body of inferential, computational, and stochastic process theory [borrowed from the general theory of discrete exponential families, see, e.g., Barndorff-Nielsen (1978); Brown (1986)] that can be brought to bear on models specified in its terms.
We begin with the "static" case in which we have a single random graph, G, with support G. It is convenient to model G via its adjacency matrix Y, with y representing the associated support (i.e., the set of adjacency matrices corresponding to all elements in G). In ERGM form, we express the pmf of Y as follows:
where is a vector of sufficient statistics, ? s is a vector of natural parameters, X X is a collection of covariates, and y is the indicator function (i.e., 1 if its argument is in the support of y, 0 otherwise).1 If |G| is finite, then the pmf for any G can obviously be written with finite-dimensional S, ? (e.g., by letting S be a vector of indicator variables for elements of y); this is not necessarily true in the more general case, although a representation with S, ? of countable dimension still exists. In practice, it is generally assumed that S is of low dimension, or that at least that the vector of natural parameters can be mapped to a low-dimensional vector of "curved" parameters [see, e.g., Hunter and Handcock (2006)].
While the extreme generality of this framework has made it attractive, model selection and parameter estimation are often difficult due to the normalizing factor (?(?, S, X) = Sy´y exp(?T S(y´, X))) in the denominator of equation (1.1). This normalizing factor is analytically intractable and difficult to compute, except in special cases such as the Bernoulli and dyad-multinomial random graph families (Holland and Leinhardt, 1981); the first applications of this family (stemming from Holland and Leinhardt's seminal 1981 paper) focused on these special cases. Later, Frank and Strauss (1986) introduced a more general estimation procedure based on cumulant methods, but this proved too unstable for practical use. This, in turn, led to an emphasis on approximate inference using maximum pseudo-likelihood (MPLE) estimation (Besag, 1974), as popularized in this application by Strauss and Ikeda (1990) and later Wasserman and Pattison (1996). Although MPLE coincides with maximum likelihood estimation (MLE) in the limiting case of edgewise independence, the former was found to be a poor approximation to the MLE in many practical settings, thus leading to a consensus against its general use [see, e.g., Besag (2001) and van Duijn et al. (2009)]. The late 1990s saw the development of effective Markov chain Monte Carlo strategies for simulating draws from ERG models (Anderson et al., 1999; Snijders, 2002) which led to the current focus on MLE methods based either on first order method of moments (which coincides with MLE for regular ERGMs) or on importance sampling (Geyer and Thompson, 1992).2
Theoretical developments in the ERGM literature have...
