Session 1A: Overview John Geweke Bayesian 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, 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.
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)
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.
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
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
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
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
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.
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
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).
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.
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.
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
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
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
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?