Session 1A: Overview

Transcription

1 Session 1A: Overview John Geweke Bayesian Econometrics and its Applicatoins August 13, 2012

2 Motivation Motivating examples Drug testing and approval Climate change Mergers and acquisition Oil re ning

3 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, and the consequences of the decision, are quantitative. The relationship between information and consequences is not deterministic. 4 Multiple sources of information bear on the decision.

4 Motivation Investigators and clients Investigator: Econometrician who conveys quantitative information so as to facilitate and thereby improve decisions Client Actual decision-maker (known) Another scientist (known or anonymous) A reader of a paper (anonymous)

5 Motivation Communicating e ectively with clients 1 Make all assumptions explicit. 2 Explicitly quantify all of the essentials, including the assumptions. 3 Synthesize, or provide the means to synthesize, di erent approaches and models. 4 Represent the inevitable uncertainty in ways that will be useful to the client.

6 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 xed De 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

7 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 can be improved. Well-known example: Newtonian physics Works ne in sending people to the moon. Doesn t work so ne using an electronic navigation system to drive a few kilometers

8 Observables, unobservables and objects of interest 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 risk simple model

9 Observables, unobservables and objects of interest Information about unobservables Representing what we know about θ: p (θ) Then, formally, Z p (y) = p (θ) p (y j θ) dθ. This is potentially useful. Important part of our technical work this week: How we obtain information about θ How p (θ) changes in response to new information

10 Observables, unobservables and objects of interest Conditioning on a model We have been implicitly conditioning on a model. Let s make this explicit: p (y j θ A, A) p (θ A j A) θ A 2 Θ A R k A Di erent models lead to di erent conclusions. This week, we shall see how to avoid conditioning on a particular model. The overriding principle: Use distributions of the things you don t know conditional on the things you do know.

11 Observables, unobservables and objects of interest 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, θ A, A) Otherwise, it can t be used for the decision at hand. Example: ω : 5 1, value of the portfolio at the close of the next 5 business days

12 Observables, unobservables and objects of interest A complete model A Three components: p (θ A j A) p (y j θ A, A) p (ω j y, θ A, A) Implies the joint probability density p (θ A, y, ω j A) = p (θ A j A) p (y j θ A, A) p (ω j y, θ A, A).

13 Conditioning and updating Ex ante and ex post A critical distinction Before we observe the observable, y, it is random After we observe the observable it is xed. To preserve this distinction y: ex ante y o : ex post Implication: the relevant probability density for a decision based on the model A is p (ω j y o, A) This is the single most important principle in Bayesian inference in support of decision making.

14 Conditioning and updating Details and notation Prior density: Observables density: p (θ A j A) p (y j θ A, A) The distribution of the unobservable θ A, conditional on the observed y o, has density p (θ A j y o, A) = p (θ A, y o j A) p (y o = p (θ A j A) p (y o j θ A, A) j A) p (y o j A) p (θ A j A) p (y o j θ A, A). This is the posterior density of θ A.

15 Conditioning and updating Being explicit about time For t = 0,..., T de ne Yt 0 = y1, 0..., yt 0 Then where Y 0 = f?g p (y j θ A, A) = T t=1 p (y t j Y t 1, θ A, A). This forward recursion is the way we construct dynamic models in economics. A generalization of time in this context: Information

16 Conditioning and updating Suppose Yt o0 not. Then Bayesian updating = (y1 o0,..., yo0 t ) is available, but p (θ A j Y o t, A) _ p (θ A j A) p (Y o t j θ A, A) = p (θ A j A) When yt+1 o becomes available, then t s=1 p (θ A j Yt+1, o t+1 A) _ p (θ A j A) s=1 y o0 t+1,..., yo0 T is p (y o s j Y o s 1, θ A, A). p (y o s j Y o s 1, θ A, A) _ p (θ A j Y o t, A) p (y o t+1 j Y o t, θ A, A). The concepts of prior (ex ante) and posterior (ex post) are relative, not absolute. Bayesian updating changes prior into posterior Example: August 13, 2013 closing value of the S&P 500 index

17 Conditioning and updating Concluding our rst session The probability density relevant for decision making is Z p (ω j y o, A) = p (θ A j y o, A) p (ω j θ A, y o, A) dθ A. Θ A If you ve only seen non-bayesian econometrics, this is really di erent. Likelihood-based non-bayesian statistics conditions on A and θ A, and compares the implication p (y j θ A, A) with y o. This avoids the need for any statement about the prior density p (θ A j A), at the cost of conditioning on what is unknown. Bayesian statistics conditions on y o, and utilizes the full density p (θ A, y, ω j A) to build up coherent tools for decision making, but demands speci cation of p (θ A j A). The conditioning in Bayesian statistics is driven by the actual availability of information, fully integrated with economic dynamic theory

18 Conditioning and updating 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 in econometrics (e.g. R, Stata, SAS,...) 7 Speci cally: Have you used Matlab at all?