The document discusses Bayesian econometrics, which applies Bayesian principles to economic modeling. Bayesian inference uses Bayes' theorem to update probabilities as new information becomes available. Bayesian econometrics assumes model coefficients have prior distributions and combines these priors with the sample data likelihood through Bayes' rule to obtain posterior distributions of parameters. This allows incorporating uncertain prior information not included in the model. The document provides examples of applying Bayesian methods to analyze stock market returns and informative priors used in financial economics research.

1. BAYESIAN METHOD
IN ECINOMETRICS
BU SHOKARIMOV KHOJAKBAR

2. B a ye s i a n i n f e re n c e i s a m e t h o d of s t a t i s t i c a l
i n f e re n c e i n w h i c h ba ye s t h e o re m i s u s e d t o
u p d a t e t h e p ro ba b i l i t y f o r a h y p o t h e s i s a s
i n f o r m a t i o n b e c o m e s a va i l a b l e . B a ye s i a n
i n f e re n c e i s a n i m p o r t a n t t e c h n i q u e i n
s t a t i s t i c s a n d e s p e c i a l l y i n m a t h e m a t i c s
s t a t i s t i c s . B a ye s i a n i n f e re n c e h a s f o u n d
a p p l i c a t i o n i n a w i d e r a n g e of a c t i v i t i e s ,
i n c l u d i n g s c i e n c e , e n g i n e e r i n g m e d i c i n e
s p o r t a n d l a w. I n t h e p h i l o s o p h y of d e c i s i o n
t h e o r y, B a ye s i a n i n f e re n c e i s c l o s e l y re l a t e d
t o s u b j e c t i v e p ro ba b i l i t y of t e n c a l l e d
B a ye s i a n p ro ba b i l i t y
6/7/2023

3. The ideas underlying Bayesian
statistics were developed by rev
Thomas Bayes during the 18th
century and later expanded by
Pierre Simon Laplace . As early as
1950 the potential of the Bayesian
inference in econometrics was
recognized by Jacob Marschak .
The Bayesian approach was first
applied to econometrics in the
early 1960s by W.D. Fisher , Jacques
Dreze, Clifford Hildreth . The
central motivation behind these
early endeavors in Bayesian
econometrics was the combination
of the parameter estimators with
available uncertain information on
the model parameters that was not
included in a given model
formulation
6/7/2023

4. Bayesian econometrics is a branch of econometrics which
applies Bayesian principles to economic modelling.
Bayesianism is based on a degree of belief interpretation of
probability as opposed to a relative –frequency interpretation .
The Bayesian principle relies on bayes theorem which states
that probability of B conditional on A is the ratio of joint
probability of A and divided by probability of B. Bayesian
econometrics assume that coefficients in the model have prior
distributions .
6/7/2023

5. Bayes Rule
Let x and 𝑦 be two random variables
Let 𝑃 𝑥 and 𝑃 𝑦 be the two marginal probability distribution functions of x and y
Let 𝑃 𝑥 𝑦 and 𝑃 𝑦 𝑥 denote the corresponding conditional pdfs
Let 𝑃 𝑥, 𝑦 denote the joint pdf of x and 𝑦
It is known from the law of total probability that the joint pdf can be decomposed as
𝑃 𝑥, 𝑦 = 𝑃 𝑥 𝑃 𝑦 𝑥 = 𝑃 𝑦 𝑃 𝑥 𝑦
Therefore
𝑃 𝑦 𝑥 =
𝑃 𝑦 𝑃 𝑥 𝑦
𝑃 𝑥
= 𝑐𝑃 𝑦 𝑝 𝑥 𝑦
where c is the constant of integration (see next page)
The Bayes Rule is described by the following proportion
𝑃 𝑦 𝑥 ∝ 𝑃 𝑦 𝑃 𝑥 ?
6/7/2023

6. Bayes Rule
. The essence of Bayesian econometrics is the Bayes Rule
. Ingredients of Bayesian econometrics are parameters underlying a give
model , the sample data the prior density of the parameters , the
likelihood function describing the data and the posterior distribution of
the parameters .
. A predictive distribution could also be involved .
. In the Bayesian setup , parameters are stochastic while in the classical
(non Bayesian) approach parameters are unknown constants.
. Decision making is based on the posterior distribution of the predictive
distribution of next period as described below.
6/7/2023

7. Bayes Econometrics in Financial Economics
You observe the returns on the market index over T months: 𝑟1, … , 𝑟𝑇
Let 𝑅: 𝑟1, … , 𝑟𝑇 ’ denote the 𝑇 × 1 vector of all return realizations
Assume that 𝑟𝑡~𝑁 𝜇, 𝜎02
for 𝑡 = 1, … , 𝑇
where
µ is a stochastic random variable denoting the mean return
𝜎02
is the variance which, at this stage, is assumed to be a known constant
and returns are IID (independently and identically distributed) through time.
By Bayes rule
𝑃 𝜇 𝑅, 𝜎02 ∝ 𝑃 𝜇 𝑃 𝑅 𝜇, 𝜎02
where
𝑃 𝜇 𝑅, 𝜎02
is the posterior distribution of µ
𝑃 𝜇 is the prior distribution of µ
and 𝑃 𝑅 𝜇, 𝜎02
is the joint likelihood of all return realizations
6/7/2023

8. Bayes Econometrics: Likelihood
The likelihood function of a normally distributed return realization is given by
𝑃 𝑟𝑡 𝜇, 𝜎02 =
1
2𝜋
𝜎02
𝑒𝑥𝑝 −
1
2𝜎
0
2 𝑟𝑡 − 𝜇 2
Since returns are assumed to be IID, the joint likelihood of all realized returns is
𝑃 𝑅 𝜇, 𝜎02 = 2𝜋𝜎02 −𝑇
2𝑒𝑥𝑝 −
1
2𝜎
0
2
𝑡=1
𝑇
𝑟𝑡 − 𝜇 2
Notice:
𝑟𝑡 − 𝜇 2 = 𝑟𝑡 −
𝜇
+
𝜇
− 𝜇 2
= ν𝑠2 + 𝑇 𝜇 −
𝜇
2
since the cross product is zero, and
ν = 𝑇 − 1
𝑠2 =
1
𝑇 − 1 𝑟𝑡 −
𝜇
2
𝜇
= 1
𝑇 𝑟𝑡
6/7/2023

9. Prior
The prior is specified by the researcher based on economic theory, past experience, past data,
current similar data, etc. Often, the prior is diffuse or non-informative
For the next illustration, it is assumed that 𝑃 𝜇 ∝ 𝑐, that is, the prior is diffuse, noninformative, in that it
apparently conveys no information on the parameters of interest.
I emphasize “apparently” since innocent diffuse priors could exert substantial amount of
information about quantities of interest which are non-linear functions of the parameters.
Informative priors with sound economic appeal are well perceived in financial economics.
For instance, Kandel and Stambaugh (1996), who study asset allocation when stock returns
are predictable, entertain informative prior beliefs weighted against predictability. Pastor
and Stambaugh (1999) introduce prior beliefs about expected stock returns which consider
factor model restrictions. Avramov, Cederburg, and Kvasnakova (2017) study prior beliefs
about predictive regression parameters which are disciplined by consumption based asset
pricing models including habit formation, prospect theory, and long run risk.
Computing posterior probabilities (as opposed to posterior densities) of competing models
(e.g., Avramov (2002)) necessitates the use of informative priors. Diffuse priors won’t fit
6/7/2023

10. Thank you
6/7/2023