Nonparametric Model Checks for Time Series Hira L. Koul and Winfried Stute

Presented by Xiaojun Song (UC3M) October 25th, 2011

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Outline

1 Introduction

2 Asymptotic Behavior

3 ADF

4 References

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Introduction Asymptotic Behavior ADF References

Outline

1 Introduction 2 Asymptotic Behavior 3 ADF 4 References

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Introduction Asymptotic Behavior ADF References

Motivations

This paper studies the goodness-of-fit of a parametric model for a stationary Markovian time series Xi . Much of the literature studies the parametric modeling of the conditional mean function µ of Xi given Xi−1 . Assumes the existence of a parametric family M = {m(·, θ) : θ ∈ Θ} of functions. Estimation of θ and/or testing the hypothesis µ ∈ M . Statistical inference based on a model M should be accompanied by a proper model check.

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Procedures Let ψ be a nondecreasing real-valued function such that E|ψ(X1 − r)| < ∞, for each r ∈ R. Define the ψ-autoregressive function mψ by the requirement that E[ψ(X1 − mψ (X0 ))|X0 ] = 0

a.s.

and the corresponding marked empirical process, based on a sample of size n + 1, by Vn,ψ (x) = n−1/2

n X

ψ(Xi − mψ (Xi−1 ))1(Xi−1 ≤ x),

x ∈ R.

i=1

The process Vn,ψ takes its value in the Skorokhod space D(−∞, ∞). Extended it continuously to ±∞ by putting Vn,ψ (−∞) = 0

and Vn,ψ = n−1/2

n X

ψ(Xi − mψ (Xi−1 )).

i=1 ,

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Some Remarks

If ψ(x) = x, then mψ = µ. If ψ(x) = 1(x > 0) − (1 − α), then mψ (x) = mα (x). The choice of ψ is up to the practitioner. For example, a bounded ψ to control the outliers in the innovations.

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Composite Parametric Hypothesis

Consider the null hypothesis H0 : mψ (·) = m(·, θ0 ) for some

θ0 ∈ Θ.

Let θn be a consistent estimator of θ0 under H0 based on {Xi , 0 ≤ i ≤ n}. Define 1 Vn,ψ (x) = n−1/2

n X

ψ(Xi − m(Xi−1 , θn ))1(Xi−1 ≤ x),

x ∈ R.

i=1

Tests for H0 can be based on an appropriately scaled discrepancy of this process. For example, an analogue of the K-S test would reject H0 in −1 1 favor of H1 if sup{σn,ψ |Vn,ψ (x)| : x ∈ R} is too large, where Pn 2 −1 2 σn,ψ = n ψ (X − m(X i i−1 , θn )). i=1

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Composite Parametric Hypothesis cont.

Regularity conditions: (A1) The estimator θn satisfies n1/2 (θn − θ0 ) = n−1/2

n X

l(Xi−1 , Xi , θ0 ) + op (1)

i=1

for some q-vector valued function l such that E[l(X0 , X1 , θ0 )|X0 ] = 0 and L(θ0 ) = E[l(X0 , X1 , θ0 )lT (X0 , X1 , θ0 )] exists and is positive definite. (A2) There exists a function m ˙ from R × Θ to Rq such that m(·, ˙ θ0 ) is measurable and satisfies the following: for all k < ∞, √ sup n|m(Xi−1 , t)−m(Xi−1 , θ0 )−(t−θ0 )T m(X ˙ i−1 , θ0 )| = op (1) √

1≤i≤n, nkt−θ0 k≤k

2 and Ekm(X ˙ 0 , θ0 )k < ∞.

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More Notations

Define M˙ (x, θ0 ) = (M˙ 1 (x, θ0 ), · · · , M˙ q (x, θ0 ))T , ˙ Γ(x, θ0 ) = (Γ˙ 1 (x, θ0 ), · · · , Γ˙ q (x, θ0 ))T , with ˙ 1 − m(X0 , θ0 ))1(X0 ≤ x)], M˙ j (x, θ0 ) = E[m ˙ j (X0 , θ0 )ψ(X Z Γ˙ j (x, θ0 ) = E[m ˙ j (X0 , θ0 ) fX0 dψ1(X0 ≤ x)], 1 ≤ j ≤ q,

x ∈ R.

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Outline

1 Introduction 2 Asymptotic Behavior 3 ADF 4 References

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Theorem All finite-dimensional distributions of Vn,ψ converge weakly to those of a centered continuous Gaussian process Vψ with the covariance function Kψ (x, y) = E[ψ 2 (X1 − mψ (X0 ))1(X0 ≤ x ∧ y)]. (i) Suppose, in addition, that for some η > 0, δ > 0, (a) Eψ 4 (ε1 ) < ∞, (b) Eψ 4 (ε1 )|X0 |1+η < ∞, (c) E[ψ 2 (ε2 )ψ 2 (ε1 )|X1 |1+δ ] < ∞ and that the family of d.f.’s {Fy , y ∈ R} have Lebesgue densities {fy , y ∈ R} that are uniformly bounded, supx,y fy (x) < ∞. Then Vn,ψ ⇒ Vψ

in

D[−∞, ∞].

(1)

(ii) Suppose that ψ is bounded and the family of d.f.’s {Fy , y ∈ R} have densities {fy , y ∈ R} satisfying RLebesgue 1+δ [E{fX (x − mψ (X0 ))}]1/(1+δ) dx < ∞, for some δ > 0. Then also 0 (1) holds.

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Theorem (a) If a smooth ψ, then n X 1 T −1/2 sup Vn,ψ (x) − Vn,ψ (x) + M˙ (x, θ0 )n l(Xi−1 , Xi , θ0 ) = op (1). x∈R i=1

(b) If nonsmooth ψ, then n X 1 sup Vn,ψ (x) − Vn,ψ (x) + Γ˙ T (x, θ0 )n−1/2 l(Xi−1 , Xi , θ0 ) = op (1). x∈R i=1

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Corollary Under certain assumptions, for the smooth ψ-case, 1 Vn,ψ ⇒ Vψ1

in

D[−∞, ∞],

(2)

where Vψ1 is a centered continuous Gaussian process with the covariance function Kψ1 (x, y) = Kψ (x, y) + M˙ T (x, θ0 )L(θ0 )M˙ (y, θ0 ) − M˙ T (x, θ0 )E[1(X0 ≤ y)ψ(X1 − m(X0 , θ0 ))l(X0 , X1 , θ0 )] − M˙ T (y, θ0 )E[1(X0 ≤ x)ψ(X1 − m(X0 , θ0 ))l(X0 , X1 , θ0 )].

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One Special Structure

Recall σψ2 (x) = E[ψ 2 (ε1 )|X0 = x] and let ˙ 1 − m(X0 , θ0 ))|X0 = x] for smooth ψ and γψ (x) = RE[ψ(X γψ (x) = fx (y)ψ( dy) for nonsmooth ψ. Assume that σψ2 (x) = σψ2 a positive constant in x a.s., γψ (x) = γψ a positive constant in x a.s., and that θn satisfies (A1) with l(x, y, θ0 ) = γψ−1 Σ−1 ˙ θ0 )ψ(y − m(x, θ0 )), θ0 m(x,

x, y ∈ R,

where Σθ0 = E[m(X ˙ ˙ T (X0 , θ0 )] so that L(θ0 ) = τ Σ−1 0 , θ0 )m θ0 , with 2 2 τ = σψ /γψ . Then, Kψ1 (x, y) = E ψ 2 (X1 − m(X0 , θ0 ))[Gθ0 (x ∧ y) − ν T (x)Σ−1 θ0 ν(y)] , with ν(x) = E[m(X ˙ 0 , θ0 )1(X0 ≤ x)], x, y ∈ R.

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Outline

1 Introduction 2 Asymptotic Behavior 3 ADF 4 References

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Martingale Transformation

Write m(·) ˙ R = m(·, ˙ θ0 ), G = Gθ0 . Set A(x) = m(y) ˙ m ˙ T (y)1(y ≥ x)G( dy). Assume that A(x0 ) is nonsingular for some x0 < ∞. Then A(y) is nonsingular for all y ≤ x0 . Define Z Z x T −1 T f (x) = f (x)− m ˙ (y)A (y) m(z)1(z ˙ ≥ y)f ( dz) G( dy), x ≤ x0 . −∞

Lemma Cov[T Vψ (x), T Vψ (y)] = G(x ∧ y),

x, y ∈ R,

that is, T Vψ is a Brownian motion with respect to time G.

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Theorem Assume A(x0 ) is nonsingular for some x0 < ∞. Then, 1 sup |T Vn,ψ (x) − T Vn,ψ (x)| = op (1).

x≤x0

If in addition, (1.2), (2.1) and (2.2) hold, then T Vn,ψ ⇒ T Vψ

1 and T Vn,ψ ⇒ T Vψ

in D[−∞, x0 ].

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When T is replaced by Tn

Pn Let, for xR∈ R, Gn (x) = n−1 i=1 1(Xi−1 ≤ x) and An (x) = m(y, ˙ θn )m ˙ T (y, θn )1(y ≥ x)Gn ( dy). Define an estimator of T to be Z Z x Tn f (x) = f (x)− m ˙ T (y, θn )A−1 (y) m(z, ˙ θ )1(z ≥ y)f ( dz) Gn ( dy), n n −∞ 1 Under additional smoothness condition on m, ˙ the consistency of Tn Vn,ψ 1 for T Vn,ψ can be proved.

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Theorem Under certain assumptions, 1 sup |Tn Vn,ψ (x) − T Vn,ψ (x)| = op (1),

x≤x0

and consequently, −1 1 (·) ⇒ B ◦ G σn,ψ Tn Vn,ψ

in D[−∞, x0 ].

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Some Remarks

Suppose we are interested in testing the hypothesis H0 : mψ (x) = m(x, θ0 ) for all x ∈ [a, b] and

for some

θ0 ∈ Θ.

−1 1 By the Theorem, σn,ψ Tn Vn,ψ (·) ⇒ B ◦ G in D[a, b]. Then −1 1 1 σn,ψ [Tn Vn,ψ (·) − Tn Vn,ψ (a)] ⇒ B(G(·)) − B(G(a)) in D[a, b].

The stationarity of the increments of the Brownian motion implies that 1 1 |Tn Vn,ψ (x) − Tn Vn,ψ (a)| ⇒ sup |B(u)|. 0≤u≤1 a≤x≤b (Gn (b) − Gn (a))σn,ψ

Dn = sup

So any test of H0 based on Dn is ADF.

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Applications

For all x ≤ x0 , rewrite 1 Tn Vn,ψ (x) = n−1/2

n X i=1

[1(Xi−1 ≤ x) − n−1

n X

m ˙ T (Xj−1 , θn )

j=1

× A−1 ˙ i−1 , θn )1(Xj−1 ≤ Xi−1 ∧ x)] n (Xj−1 )m(X × ψ(Xi − m(Xi−1 , θn )). Let g1 , · · · , gq be known real-valued G-square integrable functions on R and consider the class of models M with m(x, θ) = g1 (x)θ1 + · · · + gq (x)θq .

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Outline

1 Introduction 2 Asymptotic Behavior 3 ADF 4 References

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Introduction Asymptotic Behavior ADF References

Nonparametric model checks for regression, 1997, W. Stute. Model checks for regression: An innovation process approach, 1998, W. Stute, S. Thies and L.X. Zhu. Bootstrap approximations in model checks for regression, 1998, W. Stute, W.G. Manteiga and M.P. Quindimil. Nonparametric model checks for time series, 1999, H.L. Koul and W. Stute.

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Thanks for your time!

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