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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