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GoBack EFFICIENT ESTIMATION IN PARAMETRIC AND SEMIPARAMETRIC INAR( p ) MODELS F EIKE . C. DROST, BAS J.M. WERKER and R AMON VAN DEN AKKER Tilburg University, the Netherlands departments of Econometrics & Finance F A C U L T Y O F E C O N O M

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F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 1/22

EFFICIENT ESTIMATION IN PARAMETRIC AND SEMIPARAMETRIC INAR(p) MODELS

  • FEIKE. C. DROST, BAS J.M. WERKER and RAMON VAN DEN AKKER

Tilburg University, the Netherlands departments of Econometrics & Finance

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Introduction

  • definition INAR(1)
  • definition INAR(p)
  • elementary properties
  • the problem
  • relation to literature

Parametric models Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 2/22

INAR(1) process:

INteger-valued AutoRegressive process of order 1 (AL-OSH & ALZAID (1987)) is Z+ = N ∪ {0} valued analogue of AR(1) process: Xt = θ ◦ Xt−1 + εt, t ∈ N, where θ ◦ Xt−1 is the Binomial thinning operator θ ◦ Xt−1 =

Xt−1

  • j=1

Zt

j. ■ Zt j, j ∈ N, t ∈ N iid Bernoulli(θ) ■ ε1, ε2, . . . iid with distribution G on Z+ independent of Zt j’s ◆ hence θ ◦ Xt−1 given Xt−1 Binomial(θ, Xt−1) distributed ■ starting value X0 = x0 ■ interpretation as branching process with immigration

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SLIDE 4

Introduction

  • definition INAR(1)
  • definition INAR(p)
  • elementary properties
  • the problem
  • relation to literature

Parametric models Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 3/22

INAR(p) process:

INAR(p) process is the p-lags analogue: Xt = θ1 ◦ Xt−1 + θ2 ◦ Xt−2 + · · · + θp ◦ Xt−p + εt, t ∈ N,

■ thinning operators θ1 ◦ Xt−1, . . . , θp ◦ Xt−p are independent ■ Bernoulli variables in θi ◦ Xt−i survival-probability θi

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SLIDE 5

Introduction

  • definition INAR(1)
  • definition INAR(p)
  • elementary properties
  • the problem
  • relation to literature

Parametric models Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 3/22

INAR(p) process:

INAR(p) process is the p-lags analogue: Xt = θ1 ◦ Xt−1 + θ2 ◦ Xt−2 + · · · + θp ◦ Xt−p + εt, t ∈ N,

■ thinning operators θ1 ◦ Xt−1, . . . , θp ◦ Xt−p are independent ■ Bernoulli variables in θi ◦ Xt−i survival-probability θi

Remarks:

■ we follow definition of DU & LI (1991) (standard) ■ different definition than original by AL-OSH & ALZAID (1990) ■ THIS TALK: p = 1

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SLIDE 6

Introduction

  • definition INAR(1)
  • definition INAR(p)
  • elementary properties
  • the problem
  • relation to literature

Parametric models Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 4/22

Elementary properties:

■ first two (conditional) moments: ◆ Eθ,G [Xt | Xt−1] = µε + θXt−1 ◆ varθ,G [Xt | Xt−1] = σ2 ε + θ(1 − θ)Xt−1 ◆ same autocorrelation structure as AR(p)

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SLIDE 7

Introduction

  • definition INAR(1)
  • definition INAR(p)
  • elementary properties
  • the problem
  • relation to literature

Parametric models Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 4/22

Elementary properties:

■ first two (conditional) moments: ◆ Eθ,G [Xt | Xt−1] = µε + θXt−1 ◆ varθ,G [Xt | Xt−1] = σ2 ε + θ(1 − θ)Xt−1 ◆ same autocorrelation structure as AR(p) ■ X is a Markov chain with transition probabilities

P θ,G

xt−1,xt = (Binomialθ,xt−1 ∗G)(xt)

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SLIDE 8

Introduction

  • definition INAR(1)
  • definition INAR(p)
  • elementary properties
  • the problem
  • relation to literature

Parametric models Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 4/22

Elementary properties:

■ first two (conditional) moments: ◆ Eθ,G [Xt | Xt−1] = µε + θXt−1 ◆ varθ,G [Xt | Xt−1] = σ2 ε + θ(1 − θ)Xt−1 ◆ same autocorrelation structure as AR(p) ■ X is a Markov chain with transition probabilities

P θ,G

xt−1,xt = (Binomialθ,xt−1 ∗G)(xt) ■ stationary distribution, νθ,G exists if 0 ≤ θ < 1 and µG < ∞ ◆ for p = 1 well-known ◆ in general: no explicit formula for νθ,G ◆ if Eεk 1 < ∞ then Eνθ,GXk 0 < ∞ for k = 1, 2, 3 ◆ existence facilitates asymptotic analysis

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SLIDE 9

Introduction

  • definition INAR(1)
  • definition INAR(p)
  • elementary properties
  • the problem
  • relation to literature

Parametric models Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 5/22

The problem:

Given is

■ Parametric model:

G known or belongs to smooth parametric model

■ Semiparametric model:

G unknown Goal: given observations X0, . . . , Xn estimate the parameters in the model efficiently

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SLIDE 10

Introduction

  • definition INAR(1)
  • definition INAR(p)
  • elementary properties
  • the problem
  • relation to literature

Parametric models Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 6/22

Relation to previous work:

■ estimation of θ: ◆ in parametric model:

■ FRANKE & SELIGMANN (1993): ML (only p = 1,

limit-distribution derived but no efficiency proof)

■ BRÄNNÄS & HALL (2001): GMM

◆ in semiparametric model:

■ DU & LI (1991): OLS ■ SILVA & OLIVEIRA (2005): spectral based

■ estimation of G: ◆ even inefficient estimation of G not considered before

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SLIDE 11

Introduction Parametric models

  • the model
  • efficient estimation (1)
  • efficient estimation (2)
  • efficient estimation (3)

Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 7/22

Parametric model:

The model:

■ θ ∈ (0, 1) ■ G ∈ GA = (Gα|α ∈ A) ◆ Gα has finite third moment ◆ A ⊂ Rq open and convex ◆ smoothness conditions on α → Gα

Goal: Given observations X0, . . . , Xn estimate (θ, α) efficiently THIS TALK: G known

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Introduction Parametric models

  • the model
  • efficient estimation (1)
  • efficient estimation (2)
  • efficient estimation (3)

Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 8/22

Model is Locally Asymptotically Normal:

Model has LAN property, i.e. for u ∈ R: dP(n)

θ+u/√n

dP(n)

θ

(X0, . . . , Xn) = exp

  • uSn(θ) − u2

2 Iθ + oPν,θ,G(1)

  • ,

where the score Sn(θ)

d

− → N(0, Iθ) under Pθ

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Introduction Parametric models

  • the model
  • efficient estimation (1)
  • efficient estimation (2)
  • efficient estimation (3)

Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 8/22

Model is Locally Asymptotically Normal:

Model has LAN property, i.e. for u ∈ R: dP(n)

θ+u/√n

dP(n)

θ

(X0, . . . , Xn) = exp

  • uSn(θ) − u2

2 Iθ + oPν,θ,G(1)

  • ,

where the score Sn(θ)

d

− → N(0, Iθ) under Pθ This makes life tractable: (d/ dθ) log P θ,G

Xt−1,Xt = Eθ[ ˙

sθ,Xt−1(θ ◦ Xt−1) | Xt, Xt−1], where ˙ sθ,Xt−1(·) is score of Binomial(θ, Xt−1)

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Introduction Parametric models

  • the model
  • efficient estimation (1)
  • efficient estimation (2)
  • efficient estimation (3)

Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 8/22

Model is Locally Asymptotically Normal:

Model has LAN property, i.e. for u ∈ R: dP(n)

θ+u/√n

dP(n)

θ

(X0, . . . , Xn) = exp

  • uSn(θ) − u2

2 Iθ + oPν,θ,G(1)

  • ,

where the score Sn(θ)

d

− → N(0, Iθ) under Pθ This makes life tractable: (d/ dθ) log P θ,G

Xt−1,Xt = Eθ[ ˙

sθ,Xt−1(θ ◦ Xt−1) | Xt, Xt−1], where ˙ sθ,Xt−1(·) is score of Binomial(θ, Xt−1) Intuition: additional observation θ ◦ Xt−1 transition-score equals ˙ sθ,Xt−1(θ ◦ Xt−1)

  • nly observe Xt−1, Xt =

⇒ loss of information = ⇒ transition-score is coarsened

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Introduction Parametric models

  • the model
  • efficient estimation (1)
  • efficient estimation (2)
  • efficient estimation (3)

Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 9/22

Recall: convolution-theorem

■ the LAN-property (LE CAM) means that locally the estimation

  • f θ corresponds to estimation of u from one observation

Y ∼ N(u, I−1

θ )

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SLIDE 16

Introduction Parametric models

  • the model
  • efficient estimation (1)
  • efficient estimation (2)
  • efficient estimation (3)

Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 9/22

Recall: convolution-theorem

■ the LAN-property (LE CAM) means that locally the estimation

  • f θ corresponds to estimation of u from one observation

Y ∼ N(u, I−1

θ ) ■ for many loss functions: Y is efficient estimator of u =

⇒ by LE CAM theory this defines optimality in LAN-experiments

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SLIDE 17

Introduction Parametric models

  • the model
  • efficient estimation (1)
  • efficient estimation (2)
  • efficient estimation (3)

Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 9/22

Recall: convolution-theorem

■ the LAN-property (LE CAM) means that locally the estimation

  • f θ corresponds to estimation of u from one observation

Y ∼ N(u, I−1

θ ) ■ for many loss functions: Y is efficient estimator of u =

⇒ by LE CAM theory this defines optimality in LAN-experiments

■ recall HÁJEK-LE CAM convolution-theorem:

if ˆ θn is regular estimator of θ, i.e. √n(ˆ θn − (θ + u/√n))

dθ+u/√n

→ Zθ for all u ∈ R, then Zθ

d

= N(0, I−1

θ ) ⊕ Wθ,ˆ θn

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SLIDE 18

Introduction Parametric models

  • the model
  • efficient estimation (1)
  • efficient estimation (2)
  • efficient estimation (3)

Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 9/22

Recall: convolution-theorem

■ the LAN-property (LE CAM) means that locally the estimation

  • f θ corresponds to estimation of u from one observation

Y ∼ N(u, I−1

θ ) ■ for many loss functions: Y is efficient estimator of u =

⇒ by LE CAM theory this defines optimality in LAN-experiments

■ recall HÁJEK-LE CAM convolution-theorem:

if ˆ θn is regular estimator of θ, i.e. √n(ˆ θn − (θ + u/√n))

dθ+u/√n

→ Zθ for all u ∈ R, then Zθ

d

= N(0, I−1

θ ) ⊕ Wθ,ˆ θn ■ if ˆ

θn is regular and √n(ˆ θn − θ)

dθ

→ N(0, I−1

θ ) then ˆ

θn is called efficient

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SLIDE 19

Introduction Parametric models

  • the model
  • efficient estimation (1)
  • efficient estimation (2)
  • efficient estimation (3)

Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 10/22

Efficient estimation by one-step method:

■ use as initial (inefficient) √n- estimator of θ OLS (DU & LI

(1991))

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SLIDE 20

Introduction Parametric models

  • the model
  • efficient estimation (1)
  • efficient estimation (2)
  • efficient estimation (3)

Semiparametric model The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 10/22

Efficient estimation by one-step method:

■ use as initial (inefficient) √n- estimator of θ OLS (DU & LI

(1991))

■ then (discretizing ˆ

θn yields ˆ θ∗

n)

˜ θn = ˆ θ∗

n+ 1

√n   − 1 n

n

  • t=1

  d2 dθ2 log P θ,G

Xt−1,Xt

  • θ=ˆ

θ∗

n

 

2

 

−1

Sn(ˆ θ∗

n)

yields efficient estimator of θ

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SLIDE 21

Introduction Parametric models Semiparametric model

  • the model
  • inefficient estimation of G
  • efficient estimation
  • problem
  • solution
  • maximum likelihood
  • efficiency MLE

The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 11/22

Semiparametric model:

The model:

■ θ ∈ (0, 1) ■ G ∈ G3 = {df’s on Z+ with EGε3 1 < ∞, g(0) ∈ (0, 1)}

Goal: Given observations X0, . . . , Xn estimate (θ, G) efficiently

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SLIDE 22

Introduction Parametric models Semiparametric model

  • the model
  • inefficient estimation of G
  • efficient estimation
  • problem
  • solution
  • maximum likelihood
  • efficiency MLE

The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 12/22

Inefficient estimation of G:

In classic AR(1) model (Xt = µ + θXt−1 + εt):

■ estimate residuals by ˆ

εt = Xt − ˆ µ − ˆ θXt−1

■ estimate G by empirical df of ˆ

ε’s

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SLIDE 23

Introduction Parametric models Semiparametric model

  • the model
  • inefficient estimation of G
  • efficient estimation
  • problem
  • solution
  • maximum likelihood
  • efficiency MLE

The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 12/22

Inefficient estimation of G:

In classic AR(1) model (Xt = µ + θXt−1 + εt):

■ estimate residuals by ˆ

εt = Xt − ˆ µ − ˆ θXt−1

■ estimate G by empirical df of ˆ

ε’s In INAR(1) case:

■ given Xt−1, Xt θ ◦ Xt−1 is still random ■ note that if Xt−1 = 0 then Xt = εt ■ estimate G(k) by

ˆ Gn(k) = 1

n

  • t 1{Xt−1=0,Xt≤k}

then √n( ˆ Gn(k) − G(k))

d

− → N

  • 0,

1 νθ,G{0}G(k)(1 − G(k))

  • ■ however: turns out to be inefficient
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SLIDE 24

Introduction Parametric models Semiparametric model

  • the model
  • inefficient estimation of G
  • efficient estimation
  • problem
  • solution
  • maximum likelihood
  • efficiency MLE

The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 13/22

Review: efficient estimation

■ view Θ × G3 as subset of [0, 1] × ℓ∞(Z+); ■ class of estimators we consider are the (semi-parametric)

regular estimators

■ if (ˆ

θn, ˆ Gn) is regular with limit-distribution Z then Z = L ⊕ N where L is mean-zero Gaussian and only depends on the model and N is additional noise

■ the covariance structure of L is determined by the efficient

influence operator

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SLIDE 25

Introduction Parametric models Semiparametric model

  • the model
  • inefficient estimation of G
  • efficient estimation
  • problem
  • solution
  • maximum likelihood
  • efficiency MLE

The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 14/22

Problem:

■ normally the efficient influence operator can be quite

explicitly computed

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SLIDE 26

Introduction Parametric models Semiparametric model

  • the model
  • inefficient estimation of G
  • efficient estimation
  • problem
  • solution
  • maximum likelihood
  • efficiency MLE

The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 14/22

Problem:

■ normally the efficient influence operator can be quite

explicitly computed

■ then one proceeds: ◆ if you have estimator: check regularity estimator and

whether it attains the bound (N=0)

◆ if you have initial √n-consistent estimator: using efficient

influence operator update it into efficient estimator

slide-27
SLIDE 27

Introduction Parametric models Semiparametric model

  • the model
  • inefficient estimation of G
  • efficient estimation
  • problem
  • solution
  • maximum likelihood
  • efficiency MLE

The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 14/22

Problem:

■ normally the efficient influence operator can be quite

explicitly computed

■ then one proceeds: ◆ if you have estimator: check regularity estimator and

whether it attains the bound (N=0)

◆ if you have initial √n-consistent estimator: using efficient

influence operator update it into efficient estimator

■ for this model we cannot compute the efficient influence

  • perator

= ⇒ how to proceed?

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SLIDE 28

Introduction Parametric models Semiparametric model

  • the model
  • inefficient estimation of G
  • efficient estimation
  • problem
  • solution
  • maximum likelihood
  • efficiency MLE

The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 15/22

Solution:

■ if the model is smooth enough then MLE is often efficient in

traditional parametric models;

■ VAN DER VAART (1995) provides method to prove efficiency

  • f infinite-dimensional M-estimators without having to know

efficient influence operator

◆ we made some (easy) changes to this iid framework to

apply it in a time-series context)

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SLIDE 29

Introduction Parametric models Semiparametric model

  • the model
  • inefficient estimation of G
  • efficient estimation
  • problem
  • solution
  • maximum likelihood
  • efficiency MLE

The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 16/22

Maximum likelihood estimator:

Use (conditional) (nonparametric) maximum likelihood: (ˆ θn, ˆ Gn) ∈ argmax

θ∈[0,1],G∈G3 n

  • t=1

P θ,G

Xt−1,Xt

slide-30
SLIDE 30

Introduction Parametric models Semiparametric model

  • the model
  • inefficient estimation of G
  • efficient estimation
  • problem
  • solution
  • maximum likelihood
  • efficiency MLE

The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 16/22

Maximum likelihood estimator:

Use (conditional) (nonparametric) maximum likelihood: (ˆ θn, ˆ Gn) ∈ argmax

θ∈[0,1],G∈G3 n

  • t=1

P θ,G

Xt−1,Xt ■ maximum indeed exists (possibly non-unique) ◆ ˆ

Gn concentrates on {0, . . . , maxt≤n Xt}

◆ problem reduces to maximizing polynomial on compact

set

◆ algorithms available

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SLIDE 31

Introduction Parametric models Semiparametric model

  • the model
  • inefficient estimation of G
  • efficient estimation
  • problem
  • solution
  • maximum likelihood
  • efficiency MLE

The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 16/22

Maximum likelihood estimator:

Use (conditional) (nonparametric) maximum likelihood: (ˆ θn, ˆ Gn) ∈ argmax

θ∈[0,1],G∈G3 n

  • t=1

P θ,G

Xt−1,Xt ■ maximum indeed exists (possibly non-unique) ◆ ˆ

Gn concentrates on {0, . . . , maxt≤n Xt}

◆ problem reduces to maximizing polynomial on compact

set

◆ algorithms available ■ Consistency:

ˆ θn

p

− → θ and sup

x∈Z+

| ˆ Gn(x) − G(x)|

p

− → 0.

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SLIDE 32

Introduction Parametric models Semiparametric model

  • the model
  • inefficient estimation of G
  • efficient estimation
  • problem
  • solution
  • maximum likelihood
  • efficiency MLE

The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 17/22

Efficiency of MLE (1):

■ fix a realization ω ■ let h ∈ H, the set of real-valued bounded functions on Z+ ■ create path through (ˆ

θn(ω), ˆ Gn(ω)) in Θ × G3 by dGt =

  • 1 + t(h −
  • h d ˆ

Gn(ω))

  • d ˆ

Gn(ω) and θt = ˆ θn(ω) + t P(n)

t

denotes law of X0, . . . , Xn under (θt, Gt)

■ in the parametric model (P(n) t

| |t| < ǫ) the likelihood is maximal for t = 0 if realization is ω

■ hence

  • t

∂ ∂t log P θt,Gt

Xt−1,Xt(ω)

  • t=0

= 0

slide-33
SLIDE 33

Introduction Parametric models Semiparametric model

  • the model
  • inefficient estimation of G
  • efficient estimation
  • problem
  • solution
  • maximum likelihood
  • efficiency MLE

The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 18/22

Efficiency of MLE (2):

In this way we see that MLE solves Ψn(ˆ θn, ˆ Gn) = 0 where Ψn : [0, 1] × G3 → R × ℓ∞(H) given by Ψn1(θ, G) = 1 n

  • t

Eθ,G

  • ˙

sθ,Xt−1(θ ◦ Xt−1) | Xt, Xt−1

  • Ψn2(θ, G)h = 1

n

  • t

Eθ,G

  • h(εt) −
  • h dG | Xt, Xt−1
  • ,

h ∈ H

slide-34
SLIDE 34

Introduction Parametric models Semiparametric model

  • the model
  • inefficient estimation of G
  • efficient estimation
  • problem
  • solution
  • maximum likelihood
  • efficiency MLE

The unit root case Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 19/22

Efficiency maximum likelihood estimator (3):

We proved that

■ LAN-property along parametric submodels holds ■ Ψn has certain properties ◆ Donsker conditions on Ψn ◆ Fréchet differentiability of limit of Ψn ◆ properties of Fréchet derivative

which yields, by application of VAN DER VAART (1995), efficiency of MLE

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SLIDE 35

Introduction Parametric models Semiparametric model The unit root case

  • superefficiency MLE
  • limit experiment?

Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 20/22

Super-efficient behavior of MLE:

■ the AR(1) process has LABF-property at θ = 1 ■ what about INAR(1) at θ = 1?

Suppose θ = 1 is the truth and G = Geometric(1/2)

■ Xt = X0 + t i=1 εi ■ Xt ≥ Xt−1 ■ straightforward calculation yields

P θ

Xt−1,Xt =

1 2Xt (1 + θ)Xt−1

■ MLE: ˆ

θn = 1 = ⇒ with ML no estimation error at all

slide-36
SLIDE 36

Introduction Parametric models Semiparametric model The unit root case

  • superefficiency MLE
  • limit experiment?

Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 21/22

Limit experiment?

■ we have

dP

1− h

rn

(n)

dP1

(n)

(X0, . . . , Xn)

p1

→     

rn n2 → 0

exp

  • − h

2

  • rn

n2 → 1

1

rn n2 → ∞ ■ deterministic!? =

⇒ uniform limit experiment at θ = 1?

■ to determine limit-experiment (if any) we also need to

determine limits of dP

1− h

rn

(n)

dP

1− h0

rn

(n)

(X0, . . . , Xn) under P1− h0

rn

◆ =

⇒ current research

slide-37
SLIDE 37

Introduction Parametric models Semiparametric model The unit root case Summary

  • Summary

F A C U L T Y

O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N

Efficient estimation in INAR(p) models Debrecen, August 2005 - p. 22/22

Summary:

Conclusions:

■ efficient estimators in parametric and semiparametric models

Future research:

■ explore limit-experiment at θ = 1