高级计量经济学

Financial Econometrics George J. Jiang and Guanzhong Pan
Chapter 11
Outline Cumulant Matching, Method of Moments and GMM Maximum Likelihood Estimation Quasi-Maximum Likelihood Estimat
t =1
Financial Econometrics George J. Jiang and Guanzhong Pan
Chapter 11
Outline Cumulant Matching, Method of Moments and GMM Maximum Likelihood Estimation Quasi-Maximum Likelihood Estimat
Method of cumulant matching Cumulant generating function:
g (u) = ln(E [eiuX ]).
Cumulant:
Kn
=
1 in
g
(n)
(0).
The cumulants of normal distribution X ∼ N (µ, σ2): K1 = µ, K2 = σ2, K3 = K4 = · · · = 0.
Chapter 11
Outline Cumulant Matching, Method of Moments and GMM Maximum Likelihood Estimation Quasi-Maximum Likelihood Estimat
GMM estimation: Population moments: E [ft(θ)] = 0, where we have N moment conditions and K parameters, and N ≥ K ; Sample moments:
Outline Cumulant Matching, Method of Moments and GMM Maximum Likelihood Estimation Quasi-Maximum Likelihood Estimat
Cumulants and moments:
n−1 n − 1
Kn = mn −
Avar(θˆT ) = (G WG)−1G WSWG(G WG)−1,
where
G
≡
E
[
∂
ft (θ) ∂θ
].
Financial Econometrics George J. Jiang and Guanzhong Pan
Chapter 11
Outline Cumulant Matching, Method of Moments and GMM Maximum Likelihood Estimation Quasi-Maximum Likelihood Estimat
,
2
Cumulants of log return ∆Zt:
K1 = λµJ , K2 = σ2 + λ(µ2J + σJ2), K3 = λµJ (µ2J + 3σJ2), K4 = λ(µ4J + 6µ2J σJ2 + 3σJ4).
Financial Econometrics George J. Jiang and Guanzhong Pan
k − 1 Kk mn−k .
k =1
We have
K1 = m1, K2 = m2 − m12, K3 = m3 − 3m1m2 + 2m13, K4 = m4 − 3m22 − 4m1m3 + 12m12m2 − 6m14,
···
Cumulant matching: match sample cumulants K¯n with population cumulants Kn. cumulant matching estimator is consistent.
1T gT (θ) ≡ T ft (θ)
t =1
GMM estimator:
θˆT = arg min{JT (θ) ≡ gT (θ)W(θ)gT (θ)},
θ∈Θ
where Θ is the parameter space, W is called weighting matrix, it is a symmetric positive definite matrix and converges in probability to a positive definite matrix W, e.g.,W = I.
Example: method of moments for GBM Geometric Brownian motion:
d ln St = µ dt + σ dWt , The distribution of log return:
rt ≡ ln St∆ − ln S(t−1)∆ ∼ N (µ∆, σ2∆), Two moments:
Large sample properties of GMM estimator (Hansen, 1982): 1 Consistency: θˆT →p θ; 2 Asymptotic normality: if
√1
T
ft (θ) →d N(0, S),
T t=1
where S is a N × N positive definite matrix, then θˆT is asymptotically normal with variance-covariance matrix
Financial Econometrics George J. Jiang and Guanzhong Pan
Chapter 11
Outline Cumulant Matching, Method of Moments and GMM Maximum Likelihood Estimation Quasi-Maximum Likelihood Estimat
Chapter 11
Outline Cumulant Matching, Method of Moments and GMM Maximum Likelihood Estimation Quasi-Maximum Likelihood Estimat
Sample moments:
1 m¯ n ≡ T
Example: Estimating jump-diffusion process
Nt
Zt = Z0 + σWt + Yk .
k =1
Cumulant generating function of log return ∆Zt:
u2σ2 g (u) = − + λ
eiuµJ −u2σJ2/2 − 1
George J. Jiang and Guanzhong Pan
Financial Econometrics George J. Jiang and Guanzhong Pan
Chapter 11
Outline Cumulant Matching, Method of Moments and GMM Maximum Likelihood Estimation Quasi-Maximum Likelihood Estimat
Financial Econometrics George J. Jiang and Guanzhong Pan
Chapter 11
Outline Cumulant Matching, Method of Moments and GMM Maximum Likelihood Estimation Quasi-Maximum Likelihood Estimat
Method of Moments: Match sample moments with population moments, the number of moment conditions is equal to the number of parameters. K parameters: θ = (θ1, · · · , θK ) . K moments: E [ft(θ)] = 0 Method of moments: 1T mT (θ) ≡ T ft (θ) = 0.
Chapter 11
Outline Cumulant Matching, Method of Moments and GMM Maximum Likelihood Estimation Quasi-Maximum Likelihood Estimat
Asymptotic variance-covariance matrix:
The cumulants of Poisson distribution Y ∼ Poisson(λ): K1 = K2 = K3 = K4 = · · · = λ.
Financial Econometrics George J. Jiang and Guanzhong Pan
Chapter 11
σ2 T∆
0
0 2σ4 .
T
Replace
Leabharlann Baidu
σ2
with
its
estimator
σˆ2
=
1 T∆
T t =1
(rt
−
¯r )2,
we
get
the estimator of variance-covariance matrix.
Financial Econometrics George J. Jiang and Guanzhong Pan
T
∆Ztn,
n = 1, 2, 3, 4,
t =1
Cumulant matching:
µˆ4J
−
2K¯3 K¯1
µˆ2J
+
3 2
K¯4 K¯1
µˆJ
−
1 K¯32 2 K¯12
=
0,
(1)
λˆ = K¯1 ,
(2)
µˆJ
σˆJ2
=
K¯3
− µˆ2J 3K¯1
K¯1
,
(3)
σˆ2 = K¯2 − λˆ(µˆ2J + σˆJ2).
3 Quasi-Maximum Likelihood Estimation and Approximate-Maximum Likelihood Estimation QMLE AMLE
4 Characteristic Function Methods Characteristic Function with Its Important Properties Independently Identical Distributed (iid) Case Weakly Dependent Case
E [rt − µ∆] = 0, E [rt2 − (µ∆)2 − σ2∆] = 0.
Method of moments estimator:
1T
µˆ = T∆
rt ,
t =1
σˆ2 = 1 T∆
T
(rt − ¯r )2,
t =1
1T
¯r ≡ T
rt .
t =1
Financial Econometrics George J. Jiang and Guanzhong Pan
Outline Cumulant Matching, Method of Moments and GMM Maximum Likelihood Estimation Quasi-Maximum Likelihood Estimat
Chapter 11
Estimation Methods of Continuous Time Models Financial Econometrics
(4)
Financial Econometrics George J. Jiang and Guanzhong Pan
Chapter 11
Outline Cumulant Matching, Method of Moments and GMM Maximum Likelihood Estimation Quasi-Maximum Likelihood Estimat
Outline
1 Cumulant Matching, Method of Moments and GMM Cumulant Matching Method of Moments GMM
2 Maximum Likelihood Estimation Properties of Maximum Likelihood Estimator Solution of Transition Density