Session 1A Beamer Macroeconometrics

8/10/2019 Session 1A Beamer Macroeconometrics

http://slidepdf.com/reader/full/session-1a-beamer-macroeconometrics 1/18

Session 1A: Overview


John GewekeBayesian Econometrics and its Applicatoins

August 13, 2012




Motivation

Motivating examples

Drug testing and approval

Climate change

Mergers and acquisition

Oil re…ning




Motivation

Common features of decision-making

1 Must act on the basis of less than perfect information.

2 Must be made at a speci…ed time.

3 Important aspects of information bearing on the decision, andthe consequences of the decision, are quantitative. Therelationship between information and consequences is not

deterministic.4 Multiple sources of information bear on the decision.




Motivation

Investigators and clients

Investigator: Econometrician who conveys quantitativeinformation so as to facilitate and thereby improve decisions

Client

Actual decision-maker (known)Another scientist (known or anonymous)

A reader of a paper (anonymous)




Motivation

Communicating e¤ectively with clients

1 Make all assumptions explicit.

2 Explicitly quantify all of the essentials, including theassumptions.

3 Synthesize, or provide the means to synthesize, di¤erentapproaches and models.

4 Represent the inevitable uncertainty in ways that will be usefulto the client.

S i 1A O i




An example

An example: value at risk

p t : Market price of portfolio, close of day t

Value at risk:

Specify t = t + s , s …xedDe…ne v t ,t : P (p t p t v t ,t ) = .05

Return at risk:

y t = log (1 + r t ) = log (p t /p t 1)r t = (p t p t 1) /p t 1

(Overly) simple model:

y t iid s N

µ, σ 2

S i 1A O i




Observables, unobservables and objects of interest

Putting models in context

George Box: “All models are wrong; some are useful.”

John Geweke: “And with inspiration and perspiration they canbe improved.”

Well-known example: Newtonian physics

Works …ne in sending people to the moon.

Doesn’t work so …ne using an electronic navigation system todrive a few kilometers






A …rst pass at models (and notation)

y: a vector of observables.

θ: a vector of unobservables (think widely)Part of the model

p (y j θ)

This may restrict behavior, but is typically useless you know

nothing about θ.Examples: the gravitational constant, and the value at risksimple model






Information about unobservables

Representing what we know about θ:

p (θ)

Then, formally,

p (y) =Z

p (θ) p (y j θ) d θ.

This is potentially useful.

Important part of our technical work this week:

How we obtain information about θHow p (θ) changes in response to new information






Conditioning on a model

We have been implicitly conditioning on a model.

Let’s make this explicit:

p (y j θA , A)p (θA j A)θA 2 ΘA R

k A

Di¤erent models lead to di¤erent conclusions.

This week, we shall see how to avoid conditioning on aparticular model.

The overriding principle: Use distributions of the things youdon’t know conditional on the things you do know.






The vector of interest

ω: The vector of interest

Directly a¤ects the consequences of a decision

(We will be more precise in the next session.)

The model must specify

p (ω j y, θA , A)

Otherwise, it can’t be used for the decision at hand.Example: ω : 5 1, value of the portfolio at the close of thenext 5 business days





A complete model A

Three components:

p (θA j A)

p (y j θA , A)

p (ω j y, θA , A)

Implies the joint probability density

p (θA , y, ω j A) = p (θA j A) p (y j θA , A) p (ω j y, θA , A) .




Conditioning and updating

Ex ante and ex post

A critical distinction

Before we observe the observable, y, it is randomAfter we observe the observable it is …xed.

To preserve this distinction

y: ex anteyo : ex post

Implication: the relevant probability density for a decision

based on the model A is

p (ω j yo , A)

This is the single most important principle in Bayesian

inference in support of decision making.





Details and notation

Prior density:

p (θA j A)

Observables density:

p (y j θA , A)

The distribution of the unobservable θA, conditional on theobserved yo , has density

p (θA j yo

, A) = p (θA , yo j A)

p (yo j A) =

p (θA j A) p (yo j θA , A)

p (yo j A)

∝ p (θA j A) p (yo j θA , A) .

This is the posterior density of θA.





Being explicit about time

For t = 0, . . . , T de…ne

Y0t = y01 , ..., y0t

where Y0 = f?g

Then

p (y j θA , A) =T

∏t =1

p (yt j Yt 1 , θA , A) .

This forward recursion is the way we construct dynamicmodels in economics.

A generalization of time in this context: Information





Bayesian updating

Suppose Yo 0t = (yo 01 , ..., yo 0t ) is available, but

yo 0t +1 , ..., yo 0T

isnot. Then

p (θA j Yo t , A) _ p (θA j A) p (Yo

t j θA , A)

= p

(θ

A j A)

t

∏s =1

p (

yo

s j Yo

s 1, θ

A, A

) .

When yo t +1 becomes available, then

p (θA j Yo t +1 , A) _ p (θA j A)

t +1

∏s =1

p (yo s j Y

o s 1 , θA , A)

_ p (θA j Yo t , A) p (yo

t +1 j Yo t , θA , A) .

The concepts of prior (ex ante) and posterior (ex post) arerelative, not absolute.Bayesian updating changes prior into posterior

Example: August 13, 2013 closing value of the S&P 500 index





Concluding our …rst session

The probability density relevant for decision making is

p (ω j yo , A) =

Z ΘA

p (θA j yo

, A) p (ω j θA , yo , A) d θA .

If you’ve only seen non-Bayesian econometrics, this is really

di¤erent.Likelihood-based non-Bayesian statistics conditions on A andθA, and compares the implication p (y j θA , A) with yo .This avoids the need for any statement about the prior densityp (θA j A), at the cost of conditioning on what is unknown.

Bayesian statistics conditions on yo , and utilizes the fulldensity p (θA , y, ω j A) to build up coherent tools for decisionmaking, but demands speci…cation of p (θA j A).

The conditioning in Bayesian statistics is

driven by the actual availability of information,

fully integrated with economic dynamic theory





Bayesian updating: Practical example

1 Name and institution

2 Do you require formal evaluation of your work in this course?

3 Did you bring a laptop?

4 If so: operating system (e.g. Windows XP, Mac OS X, Linux,...)?

5 If so: does it have Matlab installed?

6 Have you used mathematical applications software ineconometrics (e.g. R, Stata, SAS, ...)

7 Speci…cally: Have you used Matlab at all?

Documents

Session 1A Beamer Macroeconometrics