Factibilidad Operativa

Here we derive the consistency and asymptotic normality of our estimators. First we intro- duce some notation, R

k(z)dz = 1, R

zk(z)dz = 0, R

zz⊤k(z)dz = µ2(k)I, and Rk(z)2dz=

R(k), where µ2(k) ̸= 0 and R(k) ̸= 0 are scalars and I is the identity matrix. For the

first stage estimator, let vt = Xt− π (Zt), for the second stage estimator let Γ∗(z1) =

Eπ (Zt)π(Zt)⊤

Z1t= z1, for the optimal weighting matrix Ωu(z) = E utut⊤

Zt= z;

for the third stage estimator Γ(z1) = E



π (Zt)Ω(Zt)−1π (Zt)⊤

The following conditions underlay the asymptotic theory of our estimators.

Assumption 4.2.1. Let (Yt, Xt,Zt, ut) be strictly stationary and α-mixing process with

α (t) = O(t−τ), where τ = (2 + δ )(1 + δ )/δ for some δ > 0. Assumption 4.2.2. For the random errors, ut, E[ut|Zt] = 0.

Assumption 4.2.3. Let f (·) and f1(·) the probability functions of z and z1and there exist a

compact set D such that infz∈Df(z) > 0. All density functions are continuously differentiable

in all their arguments and they are bounded from above and below in any point of their support.

Assumption 4.2.4. The kernel functions L(·) and K(·) are compactly supported and bounded. Besides, let |L(z) − L(z′)| ≤ C |z − z′| for all z and z′.

Assumption 4.2.5. The bandwidth matrices H1and H2are symmetric and strictly definite

positive in such a way that H1= o(H2). Moreover, each entry of the matrices tend to zero as

T → ∞ in such a way that T |H1|1/2/ log T → ∞ and T |H2|1/2→ ∞.

Assumption 4.2.6. The second order derivatives of γ1(·) . . . γdm(·) are bounded and uniformly

continuous and satisfy the Lpschitz condition. Assumption 4.2.7. Let E vec(vt)vec(vt)⊤

 Z_t = z < ∞ and E h |vec(vt)|2+δ    Zt= z i be bounded and uniformly continuous in its support for some δ , where δ was defined in assumption 4.2.1.

Assumption 4.2.8. Let the second order derivatives of β1(·) . . . βd(·) be bounded and uni-

formly continuous in any point of their support.

Assumption 4.2.9. Let, Γ∗(z1), Ω(Zt) and Γ(z1) be positive definite, continuous and invert-

ible

Assumption 4.2.10. For the same δ defined in assumption 4.2.1, let the matrices E |Xt|2+δ <

∞, E h |u_t|2+δ Z1t= z1 i and E h  π (Zt)π(Zt)⊤   2+δ  Z1t = z1 i

be bounded and uniformly continuous in their support.

Assumption 4.2.11. Let the following matrices E h  π (Zt)Ω(Zt)−1ut   2+δ  Z1t = z1 i and E h  π (Zt)Ω(Zt)−1π (Zt)⊤   2+δ  Z1t = z1 i

be bounded and uniformly continuous in their support, for the same δ defined in assumption 4.2.1.

Assumption 4.2.12. T |H2|

1 2[1+

2

4.2 Econometric model and estimation procedure

Assumption 4.2.1 is a standard assumption in the asset pricing literature; the α-mixing condition is one of the weakest mixing conditions for weakly dependent stochastic pro- cesses. Many financial time series are α-mixing, see, Cai (2002a), Carrasco and Chen (2002), Chen and Tang (2005) among others for examples. Assumption 4.2.2 imposes the endogeneity of Xt and Z1t and the exogeneity of Zt. Assumption 4.2.3 requires the

densities of Z1t and Zt to be bounded and smooth functions , this assumption and as-

sumption 4.2.4 are standard in nonparametric literature. Assumption 4.2.5 goes in the same direction as assumptions 3 and 8 in Cai et al. (2000); these assumptions are com- mon in local fitting and two-stage nonparametric estimation. Note that here we assume that LH1(·) = |H1|

−1/2_L(·H−1/2

1 ) and KH2(·) = |H2|

−1/2_K(·H−1/2

2 ) but we can also assume

L_h1(·) = h−(p+q)₁ L(·/h1) and Kh2(·) = h

−q

2 K(·/h2); it is easy to show that our results hold

with little change if we replace assumptions 4.2.5 and 4.2.12 with 4.2.13 and 4.2.14 respec- tively.

Assumption 4.2.13. The bandwidths h1 and h2 tend to zero as T → ∞ in such a way that

T h₁p+q/ log T → ∞ and T hq₂→ ∞. Besides, we also need that h1= o(h2).

Assumption 4.2.14. T hq[1+ 2 1+δ]

2 → ∞

Assumptions 4.2.6 and 4.2.8 are common in local linear fitting literature and ensure that the Taylor approximation could carry through. Assumptions 4.2.7, 4.2.9, 4.2.10 and 4.2.11 are moment conditions similar to those in Cai et al. (2015), Cai et al. (2000), Cai et al. (2006), Cai and Li (2008), Fan and Huang (2005) or Cai et al. (2017) among others. Finally, assumption 4.2.12 is necessary for the central limit theorem, note that this assumption is not restrictive (see, Cai et al. (2015) for discussion).

The following theorems establish the main result of our work.

Theorem 4.2.1. Assuming that conditions 4.2.1-4.2.7 hold, then as T → ∞ we obtain

vec( ˆπ (z)) = vec (π (z)) +1

2µ2(L)diagdmtr Hγr(z)H1
 idm+ op(tr(H1)) ,

where diag_dmtr Hγr(z)H1
 stands for the diagonal matrix of elements tr Hγr(z)H1
, for r= 1, . . . , dm, andHγr(z) is a (p + q) × (p + q) Hessian matrix of the rth component of γ(z). Theorem 4.2.2. Assuming that conditions 4.2.1-4.2.10 hold, then as T → ∞ we obtain

¯

β (z1) = β (z1) +

1

2µ2(K)diagdtr Hβr(z1)H2
 id+ op(tr(H2)) ,

where tr H_βr(z1)H2
, for r = 1, . . . , d, and Hβr(z1) is a q × q Hessian matrix of the rth component of β (z1).

Theorem 4.2.3. Assuming that conditions 4.2.1-4.2.10 hold, then as T → ∞ we obtain ˆ

Ω(z) →PΩ(z)

Theorem 4.2.4. Assuming that conditions 4.2.1-4.2.12 hold, then as T → ∞ we obtain q T|H2|1/2 ˆβ (z1) − β (z1) − b(z1)  →N 0d, R(K) f (z1)−1Γ(z1)−1
where b(z) = 1₂µ2(K)diagdtr Hβr(z1)H2
 id.

As consequence of theorems 4.2.1-4.2.4 it is easy to verify that the estimator is consistent with a coverage rate depending on T and H2but not H1as long as the condition H1= o(H2)

is satisfied. Similar to the standard nonparametric regression (Fan and Gijbels (1996)), the bias appears mainly from the second order derivative of π(·) and β (·). Indeed, the approximation errors of the functions π(·) are transmitted to the bias in estimating β (·) but are asymptotically negligible due to our regularity conditions.