• Aucun résultat trouvé

Let y1, ..., yn be a sequence of random variables drawn from a probability distribution P0 belonging to some class P. Let d be the selection vector which denotes the elements of the candidate set that are included in a particular moment condition. Hence, dj = 1 indicates that thejth element is included in the set of moments anddj = 0 indicates that

1.2. MSC and its local robustness properties 5 this element is excluded. We denote by|d| the number of elements included and byhmax

the maximum number of orthogonality conditions. Let (Wn(d)), n ∈N be a sequence of random weight matrices that converges in limit to a positive definite matrix W0(d) for each set of moments d and h :Y ×Θ× D →R|d| be the random vector of orthogonality conditions, D being defined in Assumption A.1.5. The GMM estimator of θ0 denoted by θˆn( ˆP;d) is defined by

θˆn( ˆP;d) = arg min

θ∈ΘEPˆhT(Y, θ;d)Wn(d)EPˆh(Y, θ;d),

where ˆP is the empirical distribution. Under regularity conditions, [Hansen, 1982], the GMM estimator exists, is consistent and asymptotically normally distributed at the model, for d ∈ Z0 (defined in assumption A.1.5). The asymptotic variance-covariance matrix is given by Σ0(W0(d);d) =S0EP0[∂h∂θT(Y, θ0;d)]W0(d)V0W0(d)EP0[∂θ∂hT(Y, θ0;d)]S0, where V0 =V0(d) =EP0[h(Y, θ0;d)hT(Y, θ0;d)] and

S0 =S0(W0(d);d) =nEP0[∂h∂θT(Y, θ0;d)]W0(d)EP0[∂θ∂hT(Y, θ0;d)]o−1. The corresponding functional form of the GMM estimator is given by:

θ(Pˆ ;d) = arg min

θ∈ΘEPhT(Y, θ;d)W0(d)EPh(Y, θ;d). (1.1) [Andrews, 1999] proposes a family of procedures to asymptotically select the most efficient set of moments among sets of moment conditions that provide consistent estimates of the structural parameters given a model. Let us summarize these criteria. Consider that the functional form is fixed, i.e. the parameter space remains the same for all possible sets of orthogonality conditions. For any empirical measure, ˆP satisfying assumption A.1.6, Andrews proposes to find the set of moments ˆdMSC in the classD which minimizes

MSC( ˆP;d) =nJ( ˆP;d)p(|d|)κn , (1.2)

where J( ˆP;d) =EPˆhT(Y,θˆn;d)Wn(d)EPˆh(Y,θˆn;d).

Assumption 1.2.1 (Penalty properties for MSC). 1. p(|d|) is a strictly increas-ing function.

2. κn =o(n) and κ→ ∞.

Assumption 1.2.1 is needed for MSC to be consistent. Some specific MSC based on the corresponding classical information criteria are defined by:

MSC-AIC : nJ( ˆP;d)−2(|d| −p) MSC-BIC : nJ( ˆP;d)−(|d| −p) logn

MSC-HQIC : nJ( ˆP;d)H(|d| −p) log logn, H >2.

Note that the penalty for MSC-AIC does not satisfy the assumption κn → ∞, hence it does not choose with probability one the set of moments which maximizes the number of orthogonality conditions that are asymptotically zero as n → ∞. For MSC with κn → ∞, under some regularity conditions given in [Andrews, 1999], dˆMSC( ˆP) →P d0 and√

n(ˆθn( ˆP; ˆdMSC)−θ0)→D N(0,Σ0(W0(d0);d0)), where ’→’ and ’P →’ denote convergenceD in probability and in distribution respectively. These results whend0 is uniquely defined.

If the latter is not verified the limiting behavior of MSC will be a random variable with a probability associated with each element belonging to MZ0,MZ0 defined in A.1.5.

Finally, [Andrews, 1999] also defines two testing procedures for moment selection called Downward Testing and Upward Testing. The former chooses the greatest number of moments, for which some J( ˆP;d) test does not reject for d ∈ D at a given significance level αn, starting from the set of moments that maximizes the number of orthogonality conditions. The latter starts from the set of moments that minimizes the number of overidentifying conditions and stops when J( ˆP;d) test does not reject for d ∈ D at a given significance level αn for |d| being maximal.

These results hold when the data generating process is P0. In order to investigate the behavior of the functionals

1.2. MSC and its local robustness properties 7

J(P;d) =EPhT(Y,θ;ˆ d)W0(d)EPh(Y,θ;ˆ d)

M SC(P;d) =nJ(P;d)p(|d|)κn (1.3)

outside the reference model, we define the following neighborhood of P0

Uε,n(P0) =

(

Pε,n = 1− ε

n

!

P0+ ε

nQ | Q arbitrary

)

. (1.4)

Notice that any distribution in this neighborhood is at most at distance εn (in the sense of Kolmogorov) from the reference model P0. This is a common choice in robustness investigations to formalize deviations from the reference model (see for instance,

[Ronchetti and Trojani, 2001, p. 51]), but other types of contaminations could be consid-ered.

The purpose of robust statistics is to construct statistical procedures that exhibit a stable behavior in such a neighborhood of the reference model. It is convenient to study this behavior by formalizing the statistical procedure as a functional. Then the local robustness of the statistical functional in a neighborhood like (1.4) is achieved by imposing a bound on the influence function of the functional. A bounded influence function implies a uniform bound on the possible values of any statistic in the neighborhood defined by (1.4); see [Hampel, 1974], [Hampel et al., 1986].

For instance, the influence function of the GMM functional ˆθ, defined in (1.1), is given by IF(y,θ, Pˆ 0;d) = limε→0

θ((1−ε)Pˆ 0+εδy;d)−θ(Pˆ 0;d)

ε . In our setting, for a given set of moments d∈ D, the influence function is given implicitly by:

S0Γd(I|d|⊗IF(y,θ, Pˆ 0;d))W0EP0h(Y, θ0;d) + IF(y,θ, Pˆ 0;d) = 2S0EP0[∂h∂θT(Y, θ0;d)]W0(d)EP0[h(Y, θ0;d)]−

−S0∂h∂θT(y, θ0;d)W0EP0[h(Y, θ0;d)]S0EP0[∂h∂θT(Y, θ0;d)]W0h(y, θ0;d), (1.5)

where δy is a point mass at y, Γd = EP0[∂θ∂θ2h(1)T · · ·∂θ∂θ2h(|d|)T ], h(i) is the ith element of the vectorh(·) and I|d| is the identity matrix of size|d|. From (1.5) we can draw the following conclusions.

• A necessary condition from (1.5) to achieve local robustness for both valid and invalid moments is to have bounded functions h(·, θ0;d) and ∂θ∂hT(·, θ0;d). It is in-teresting to notice that the same conditions are required in a different context to obtain second-order robustness of M-estimators; see [La Vecchia et al., 2012].

• For valid sets of moments, Z0 in assumption A.1.5 in the Appendix, we obtain the influence function for GMM estimators developed in [Ronchetti and Trojani, 2001, Eq. (14)].

• In some particular cases we can obtain an explicit form for the influence funtion for both valid and invalid moments. One example is when Γd = 0, which corresponds to linear regression and linear IV.

• The necessary conditions that lead to local robustness are not satisfied by most of the existing GMM estimators in the literature. It will be shown in section 1.7 that all linear IV estimators have an unbounded IF; see [Ronchetti and Trojani, 2001]

for more examples.

The influence function is a key element in the application of von Mises expansions to func-tionals, see [Von Mises, 1947]. The expansions provide an approximation of the behavior of the functional in a neighborhood of the ideal distribution. In the context of Moment Selection we provide the asymptotic expansion in the neighborhood (1.4) of the reference model of any MSC defined in (1.3) .

1.3. Robust Moment Selection 9