INDICADORES PARA LA EVALUACIÓN

Now we turn our attention to more recent data regarding the S&P 500 stock index considered from January, 1990, to December, 2004, see Figure 10. This period is marked by many substantial events affecting the financial markets, ranging from September 11, 2001, terrorist attacks and the war in Iraq (2003) to the crash of the technology stock-market bubble (2000–2002). For the sake of simplicity, a particular time period is again selected. The estimation results for years 2003 and 2004, where the first one represent a more volatile period (war on terrorism in Iraq) and the latter one is a less volatile period. All adaptive methods detected rather quickly a structural break at the beginning of 2003; additionally, all methods also detected a structural break in the second half of 2003, although the adaptive GARCH did so with a delay of more than 8 months. The ratios of monthly PE of all adaptive methods to the parametric GARCH are summarized on Figure 11 ( r = 0.5 and ρ = 1.5 for all methods).

In the beginning of 2003, which together with previous year 2002 corresponds to a more volatile period (see Figure 10), all adaptive methods perform as well

02/03 07/03 11/03 04/04 09/04

1

2

Ratio of L1 errors

Time

Local Constant to parametric GARCH

02/03 07/03 11/03 04/04 09/04

1

2

Ratio of L1 errors

Time

Local ARCH to parametric GARCH

02/03 07/03 11/03 04/04 09/04

1

2

Ratio of L1 errors

Time

Local GARCH to parametric GARCH

Figure 11: The ratio of the absolute prediction errors of the three pointwise adap- tive methods to the parametric GARCH for predictions one period ahead averaged over one month horizon. The S&P 500 index is considered from January, 2003 to December, 2004.

as the parametric GARCH. In the middle of year 2003, the local constant and local ARCH models are able to detect another structural change, possibly less pronounced than the one at the beginning of 2003 because of its late detection by the adaptive GARCH. Around this period, the local ARCH shortly performs worse than the parametric GARCH. From the end of 2003 and in year 2004, all adaptive methods starts to outperform the parametric GARCH, where the reduction of the PEs due to the adaptive estimation amounts to 20% on average. All pointwise adaptive estimates exhibit a short period of instability in the first months of 2004, where their performance temporarily worsens to the level of parametric GARCH. This corresponds to “uncertainty” of the adaptive methods about the length of the interval of homogeneity. After this short period, the performance of all adaptive methods is comparable, although the local constant performs overall best of all methods (closely followed by local ARCH) judged by the global PE.

in Section 6.1, it seems that the benefit of pointwise adaptive estimation is most pronounced during periods of stability that follow an unstable period (i.e., in year 2004 here) rather than during a presumably rapidly changing environment. The reason is that, despite possible inconsistency of parametric methods under change points, the adaptive methods tend to have rather large variance when the intervals of time homogeneity become very short.

7

Conclusion

In this paper, we extend the idea of adaptive pointwise estimation to more com- plex parametric models that belong to the class of autoregressive moving-average conditional-heteroscedasticity models. In the specific case of ARCH and GARCH, which represent particularly difficult case due to high data demands and depen- dence of critical values on underlying parameters, we demonstrate the use and feasibility of the proposed procedure: on the one hand, the adaptive procedure, which itself depends on a number of auxiliary parameters, is shown to be rather insensitive to the choice of these parameters, and on the other hand, it facili- tates the global selection of these parameters by means of fit or forecasting cri- teria. Further, the real-data applications highlight the flexibility of the proposed time-inhomogeneous models since even simple varying-coefficients models such as ARCH(1) can outperform standard parametric methods such as GARCH(1,1).

A

Proofs

Proof of Corollary 2.1. Given the choice of zα, it directly follows from (2.8).

Proof of Theorem 3.1. We present only the idea of the proof. The detailed proof of a similar assertion for the local constant volatility modeling can be found in Spokoiny and Chen (2007). The main fact behind the proof is that selecting the critical values of order log |IK| allows to make the probability of rejecting the null

It follows from the definition that for every τ ∈ Tk ⊂ Ik+1, the test statistic

TIk+1,τ satisfies TIk+1,τ ≤ LJ(eθJ, θ

∗_{) + L}

Jc(eθ_Jc, θ∗) with J = [τ, T ] and Jc = Ik+1\ J . By Theorem 2.1, there are some positive constants C and λ such that

Pθ∗ L_J(eθ_J, θ∗) > z

≤ Ce−λz _{for any z > 0 ; an analogous inequality holds also}

for LJc(eθ

Jc, θ∗) . Therefore,

Pθ∗ T_Ik+1,τ > 2z

≤ 2Ce−λz, which obviously implies

Pθ∗ T_I

k+1,Tk > 2z

≤ 2C|Tk|e−λz.

Selecting zk in the form (3.4) then ensures that the probability of rejecting the

null hypothesis at step k is smaller than Const.(e−λa1 _{+ |I}

K/Ik|−λa2 + e−λa3) for

some sufficiently large a1, a2, a3.

Proof of Theorem 4.1. The proof is based on the following general result.

Lemma A.1. Let P and P0 be two measures such that the Kullback-leibler

divergence E log(dP /dP0) , satisfies E log(dP /dP0) ≤ ∆ < ∞. Then for any

random variable ζ with E0ζ < ∞ , it holds that

E_{log 1 + ζ
≤ ∆ + E0ζ.}

Proof. By simple algebra one can check that for any fixed y the maximum of the function f (x) = xy − x log x + x is attained at x = ey _{leading to the inequality}

xy ≤ x log x − x + ey_{. Using this inequality and the representation E log 1 + ζ
=}

E₀_{Z log 1 + ζ
with Z = dP /dP0 we obtain}

E_{log 1 + ζ
= E0}_{Z log 1 + ζ}
≤ E0 Z log Z − Z
+ E0(1 + ζ) = E0 Z log Z
+ E0ζ − E0Z + 1.

It remains to note that E0Z = 1 and E0 Z log Z

We now apply this lemma with ζ = ̺(bθ_{, θ)/R(b}θ_{, θ) and utilize that E0ζ =} E_θ̺(bθ_{, θ)/R(b}θ_{, θ) = 1 . This yields}

Eθ Z_I,θlog Z_I,θ

= E log ZI,θ= E X t∈I log p[Yt, g(Xt)] p[Yt, g(Xt(θ))] = EX t∈I En_log p[Yt, g(Xt)] p[Yt, g(Xt(θ))]   Ft−1 o = E∆I(θ). 

Proof of Corollary 4.1. It is Theorem 4.1 formulated for ̺(θ′, θ) = LI(θ′, θ) .

Proof of Theorem 4.2. The first inequality follows from Corollary 4.1, the second one from condition (3.3) and the property x ≥ log x for x > 0.

Proof of Theorem 4.3. By assumption bk > l , the interval Il+1 is accepted by

our multiple testing procedure. This particularly means that the sup-LR statistic TIl+1 does not exceed zl+1, and therefore for τ = tl, it holds TIl+1,tl ≤ zl+1. With Jl= Il+1\ Il, this means that

TIl+1,tl = LIl(eθIl) + LJl(eθJl) − LIl+1(eθIl+1)

= LIl(eθIl, eθIl+1) + LJl(eθJl, eθIl+1) ≥ LIl(eθIl, eθIl+1). Therefore, TIl+1 ≤ zl+1 implies LIl(eθIl, eθIl+1) ≤ zl+1 as required.