Non-parametric estimation of a time varying GARCH model

Further, if E|vt| 8 < ∞, then the asymptotic variance of the estimator is
V ar(ˆα0(u0),..., ˆαp(u0))
= 1
nh1 e⊤ 1,d+1A −1
1 B1A −1
1 e1,d+1V ar(v 2 t )S −1 ΩS −1 (1 + oP(1)),
Interestingly, the bias expression for ˆαj(u0) depends on the (d+1) th derivative of αj(u0)
only due to the structure of the model. The procedure introduced in Step 1 can be
used for the estimation of a time varying ARCH (p) model. Now it is clear that the
MSE of the estimator ˆαj(u0) is OP(h 2d+2
1
+ (nh1) −1 ). Also, when the optimal bandwidth
h1 = O(n −1/(2d+3) ) is used, then the local polynomial estimator achieves the optimal rate
of convergence OP(n −(2d+2)/(2d+3) ) for estimating αj(u0). Notice that for d = 3, the opti-
mal convergence rate is OP(n −8/9 ). Now in the following corollary, we show the asymptotic
normality of the estimator as a simple application of the martingale central limit theorem.
Corollary 4.1. Under the same assumptions as that of Theorem 4.1,
√
nh1 (ˆαtvARCH(u0) − αtvARCH(u0) − b(u0)) D �
→
Np+1 0,e ⊤ 1,d+1A −1
1 B1A −1
1 e1,d+1V ar(v2 t )S−1ΩS −1�
where b(u0) = Bias(ˆαtvARCH(u0)) and D → denotes the convergence in distribution.
Corollary 4.2. Let ˆσ 2 t = ˆαtvARCH(ut) ⊤ [1,ǫ2 t−1,...,ǫ 2 t−p] ⊤ (p+1)×1 . Then under the Assump-
tions 1 and 2,
where 0 < ρ < 1 and pn → ∞ as n → ∞.
Bias(ˆσ 2 t ) = E(ˆσ 2 t − σ 2 t ) = OP(h d+1
1 ) + O(ρ pn )
Corollary 4.2 can be proved using Proposition 2.2, equation (5) and Theorem 4.1. It
shows that the choice of pn will contribute towards the bias of the conditional variance
in the initial step by a term which decays geometrically. Therefore, this term will have
negligible effect on final estimators as pn → ∞. In Theorem 4.2, we derive the asymp-
totic bias and the variance of the estimators of tvGARCH parameter functions obtained
in Step 2. Towards this, first we introduce few more notations.
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