Mostly Harmless Bayesian
Econometrics
Dale J. Poirier
University of California, Irvine
November 29, 2018

2Abstract
Think back to those early years when you took your first statistics course. You studied
examples of location / scale models (i.e., the pdf is of the form p[σ -1(y - μ) ], where μ 0 U and σ 0U+). Frequentists developed simple estimators with known ex ante finite sampling distributions
leading in many cases to “optimal” procedures for point estimation, interval estimation, hypothesis
testing, and prediction. Likewise Bayesians developed corresponding ex post finite posterior
distributions, optimal point estimates, highest posterior density (HPD) regions, Bayes factors, andpredictive distributions. Conjugate priors were readily available (when sampling from theexponential family) and integrals had analytical closed-form expressions. In short, the models weresimple and there were a small number of easily interpretable parameters.
Now consider your last graduate course in econometrics. The models were much more complicated, the number of variables parameters became more numerous, computational demands
increased, and life moved to the magical land of Asymptopia where everything was consistent,normal, and efficient. If only you could return to that innocent world of your youth. Here I hope totake you on a journey to your Fountain of Youth.
The index of the widely praised book by Angrist and Pischke (2009) (hereafter AP09)
contains none of the words: Bayes, Bayesian, nor likelihood. Above I have inserted “Bayesian” into
their title and adopted it as the title of this monograph. I borrow those parts of AP09 that I like, andadd to it an analysis incorporating these three words. The other parts of AP09 I ignore. This is nota book review of AP09. Instead I provide a contrasting Bayesian spin on points of agreement withAP09.
In their preface, Angrist and Pischke (2009, p. xii) say:“... that empirical work is most valuable when it uses data to answer specificcausal questions.”
A distinguishing feature of their approach Angrist and Pischke (2009, pp. xii-xiii) describe as:“Most econometrics texts appear to take econometric models very seriously.
... We take a more forgiving and less literal-minded approach.” The authors claim
a “certain lack of gravitas” and “ ... not much concern with asymptotic efficiency.
Rather, our discussion of inference is devoted mostly to the finite-sample bugaboosthat should bother practitioners.”
This willingness to entertain sub-optimal efficiency is form of “econometric satisficing.”

3
My intended audience is not econometric theorists, but rather applied econometricians, andparticularly, non-Bayesians / frequentist. I believe there are many frequentists whose statisticalpractice reflects hidden beliefs about statistical matters. I do not wish to put all frequentists into a“basket of deplorables.” That has been shown to not be a winning strategy. Instead my main goal
is to create a “bigger statistical tent” within the econometrics community by interpreting nonBayesian approaches favorably from a Bayesian standpoint. I consider only random sampling andsaturated models that reduce to simple location-scale families. I intentionally omit proofs and ignore
detailing the regularity conditions under which asymptotic results follow. These can be found in theliterature I cite.
Here is the brief summary of what follows. Through a lengthy list of quotations, Chapter 1
points out the wide agreement among economists that they don’t care about the literal truth of theirassumptions, be they behavioral or probabilistic. Section 1.2 argues “truth” amounts to a sharedperception of facts. Instead of demanding “truth” of assumptions, Section 1.3 argues a more
pragmatic issue is the sensitivity of conclusions to small changes in the assumptions. Section 1.4
tackles an old issue, instrumentalism versus realism, and argues, rather than be true or realistic,
models should be instruments for prediction. Together, Sections 1.5 and 1.6 argue that the rise of
behavioral economics (Section 1.5) led to the so-called credibility revolution in econometrics(Sections 1.6). I wish to offer a Bayesian alternative to the widely popular AP09.
Chapter 2 discusses saturated models. Section 2.1 argues that all covariates are inherently
discrete, and hence, there is a positive probability of replicated observations. I focus on simple
location / scale models for clusters of observations with the same covariate values. Section 2.2
discusses an empirical example. Working with clusters within which all observations have the same
covariate values leads to exact matching of covariates. Sections 2.3 and 2.4, respectfully, introduce
important notation for univariate and multivariate models. Section 2.5 briefly discusses pooling ofclusters, a topic of great importance in subsequent material. No whimsical assumptions are neededin this chapter other than random sampling.
Chapter 3 is a general discussion of statistical inference. This is where those whimsicaldistributional assumptions arise. The likelihood approach is discussed in Section 3.2 and theBayesian approach in Section 3.3. The notion of a misspecified model (aren’t they all?) is discussed
in likelihood terms in Section 3.4 and in Bayesian terms in Section 3.5. A brief comparison of theasymptotics for frequentists and Bayesians is given in Section 3.6. For frequentists probability
reflects a unique property of reality, whereas for a Bayesians probability reflects personal beliefs
regarding reality.

4The main purpose of Chapter 3 is to find some common ground for frequentists andBayesians. This requires a likelihood and a prior. I argue that frequentists implicitly makeassumptions about sampling properties of the data when they choose their preferred estimator. In
other words, choosing an estimator presumably is made on the belief it works well for the situation
at hand. So let’s give the frequentist the benefit of doubt, and say this choice coincides with themaximum likelihood estimator (MLE). In Section 3.7 I argue this implies an implicit likelihood andconsider examples for a variety of preferred frequentist estimators. Similarly, in Section 3.8, I argue
that many other frequentist notions imply the existence of an implicit prior. For example, thecommon remark that an estimate “has the wrong sign” certainly contains beliefs that suggest wheremost of the prior’s mass should be. I argue a similar implicit prior exists when the preferred
estimator is not uniformly convergent, then the belief that the estimator’s asymptotic distribution
is sufficiently accurate in the finite sample at hand reflects an implicit prior that regions of theparameter space where the convergence is weakest are assigned small positive probability. Section
3.9 provides a few conclusions.
Chapter 4 considers the case where the endogenous variables are univariate or multivariate
normal. Different cases are considered: univariate known variances (Section 4.2), univariate
unknown homoskedastic variances (Section 4.3), univariate unknown heteroskedastic variances(Section 4.4), multivariate known variances (Section 4.5), multivariate unknown homoskedasticvariances (Section 4.6), multivariate unknown heteroskedastic variances (Section 4.7).
The argument in Chapter 2 that all data are discrete conflicts with the cases considered in
Chapter 4. So Chapter 5 considers discrete endogenous variables. It turns out that the analysis issimpler in Chapter 4. Four broad cases are considered: binomial (Section 5.2), multinomial (Section5.3), ordered multinomial (Section 5.4), and count variables (Section 5.5). Similarities tocontingency table analysis are noted.
Chapter 6 addresses the issue of randomization, which together with saturation is afundamental element of the credibility revolution. After a brief history of randomization in Section
6.1, Section 6.2 discusses frequentist criticisms and qualifications of randomization. Section 6.3
proposes a reconciliation, based on Kadane and Seidenfeld (1990), of two distinct viewpoints onwhether the purpose experiment is to learn or to prove.6
Finally, Chapter 7 offers a brief conclusion.