The “Checklist” > 2a. Estimation: Flexible Probabilities

Flexible Probabilities

The realized invariants {εt}t̄t=1 can be arranged in the realizedinformation matrix

i ≡ (ε1| · · · |εt| · · · |εt̄) ≡

ε1,1 ε1,t ε1,t̄· · ·εi,1 · · · εi,t · · · εi,t̄· · ·εı̄,1 εı̄,t εı̄,t̄

(2a.6)

How do we emphasize one joint observation versus another one?

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The “Checklist” > 2a. Estimation: Flexible Probabilities

Flexible Probabilities

Flexible Probabilities are weights

p ≡ (p1, . . . , pt̄) ≥ 0 (2a.8)

that sum to one ∑t̄t=1pt = 1 (2a.9)

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The “Checklist” > 2a. Estimation: Flexible Probabilities

Estimation

Theory

Starting from the scenario-probability pairs determined by the his-torical realizations of the invariants and the corresponding FlexibleProbabilities, estimate their distribution

{εt, pt}t̄t=1 7→ f̂ε (2a.10)

The distribution f̂ε can be obtained non-parametrically or paramet-rically and is by no means necessarily normal.

See also Practice and Intuition

More on Scenario-probability distributions

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The “Checklist” > 2a. Estimation: Flexible Probabilities

{εt}t̄t=1 invariants realizations

{εt, pt}t̄t=1

scenario-probabilityframework

εt ∼ fεdistribution

Flexible Probabilities��

Historical

++

ML

robustness

missing

observationsss

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The “Checklist” > 2a. Estimation: Flexible Probabilities

Chapter outline

The “Checklist” > 2a. Estimation: Flexible ProbabilitiesSetting the Flexible ProbabilitiesHistoricalMaximum LikelihoodGeneralized Method of MomentsRobustnessMissing dataThe final outputPoints of interest, pitfalls, practical tips

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