The “Checklist” > 2a. Estimation: Flexible Probabilities > Generalized Method of Moments

Generalized Method of Moments

• Topic: parametric estimation of the invariants distribution based onthe Generalized Method of Moments

• We introduce the Method of Moments with FlexibleProbabilities estimate

• We extend to Flexible Probabilities the Generalized Method ofMoments with exact specification

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The “Checklist” > 2a. Estimation: Flexible Probabilities > Generalized Method of MomentsMethod of Moments

Canonical Method of Moments

A parametric assumption on the distribution of the invariants is

εt ∼ fε ∈ {fθ}θ∈Θ (2a.95)

• θ ≡ (θ1, . . . , θl̄)′, l̄ × 1 vector

• Θ ⊆ Rl̄, discrete or continuous

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The “Checklist” > 2a. Estimation: Flexible Probabilities > Generalized Method of MomentsMethod of Moments

Canonical Method of Moments

The first l̄ cross moments of the invariants can be expressed by l̄ equations

fε ≡ fθε , E {εi,tεj,t · · · } ≡ hi,j,...(θε) (2a.96)

The Method of Moments (MM) estimate θ̂MM

ε of θε is the solution of

f̂MMε ≡ f

θ̂MMε

,1

t̄

∑t̄t=1εi,tεj,t · · · ≡ hi,j,...(θ̂

MM

ε ) (2a.101)

where {ε1, . . . , εt̄} is the time series of realized invariants.

See also Example 2a.24

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The “Checklist” > 2a. Estimation: Flexible Probabilities > Generalized Method of MomentsMethod of Moments

Method of Moments with Flexible Probabilities

Consider Flexible Probabilities {pt}t̄t=1 rather than the equal probabilityweights pt ≡ 1/t̄.

The Method of Moments with Flexible Probabilities (MMFP)estimate θ̂

MMFP

ε of θε is

f̂MMFPε ≡ f

θ̂MMFPε

,∑t̄

t=1ptεi,tεj,t · · · ≡ hi,j,...(θ̂MMFP

ε ) (2a.102)

See also Example 2a.25

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The “Checklist” > 2a. Estimation: Flexible Probabilities > Generalized Method of MomentsMethod of Moments

Example 2a.25. Fit of options strategy toreflected-shifted-lognormal distribution using MMFP

• Invariant: daily P&L εt ≡ Πt−1→t modeled as a reflected shiftedlognormal (2a.91)

• FP: time exponential decay probabilities, half life τHL = 500

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The “Checklist” > 2a. Estimation: Flexible Probabilities > Generalized Method of MomentsGeneralized Method of Moments - exact specification

Exact specification

Conditions on a set of l̄ arbitrary moment or orthogonality con-ditions

fε ≡ fθε , E {g (εt,θε)} ≡ 0 (2a.104)

• g (x,θ) = (g1 (x,θ) , . . . , gl̄ (x,θ))′ multivariate function

• θε ∈ Θ ⊆ Rl̄ vector of true unknown parameters

This problem is exactly specified, i.e. l̄ equations in l̄ unknowns.

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The “Checklist” > 2a. Estimation: Flexible Probabilities > Generalized Method of MomentsGeneralized Method of Moments - exact specification

Exact specification

The exact-specified Generalized Method of Moments with FlexibleProbabilities (GMMFP) estimate θ̂

GMMFP

ε of θε is the solution of

f̂GMMFPε ≡ f

θ̂GMMFPε

,∑t̄

t=1ptg(εt, θ̂GMMFP

ε ) ≡ 0 (2a.106)

Consistency: the parametric assumption (2a.95) holding true andprovided θε is the unique solution of (2a.108) (respectively, (2a.104) forexact-specified GMMFP estimate)

limens(p)→∞ θ̂GMMFP

ε = θε ⇔ limens(p)→∞ f̂GMMFPε = fε (2a.116)

where ens(p) is the Effective Number of Scenarios (2a.21).

See also Example 2a.26 and Example 2a.27

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