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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 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
INAR(1) process: INteger-valued AutoRegressive process of order 1 (A L -O SH & Introduction ● definition INAR(1) A LZAID (1987)) is Z + = N ∪ { 0 } valued analogue of AR(1) ● definition INAR( p ) ● elementary properties process: ● the problem ● relation to literature X t = θ ◦ X t − 1 + ε t , t ∈ N , Parametric models where θ ◦ X t − 1 is the Binomial thinning operator Semiparametric model The unit root case X t − 1 � Summary Z t θ ◦ X t − 1 = j . j =1 j , j ∈ N , t ∈ N iid Bernoulli( θ ) ■ Z t ■ ε 1 , ε 2 , . . . iid with distribution G on Z + independent of Z t j ’s ◆ hence θ ◦ X t − 1 given X t − 1 Binomial( θ, X t − 1 ) distributed ■ starting value X 0 = x 0 ■ interpretation as branching process with immigration 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( p ) process: INAR( p ) process is the p -lags analogue: Introduction ● definition INAR(1) ● definition INAR( p ) X t = θ 1 ◦ X t − 1 + θ 2 ◦ X t − 2 + · · · + θ p ◦ X t − p + ε t , t ∈ N , ● elementary properties ● the problem ● relation to literature ■ thinning operators θ 1 ◦ X t − 1 , . . . , θ p ◦ X t − p are independent Parametric models ■ Bernoulli variables in θ i ◦ X t − i survival-probability θ i 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: Introduction ● definition INAR(1) ● definition INAR( p ) X t = θ 1 ◦ X t − 1 + θ 2 ◦ X t − 2 + · · · + θ p ◦ X t − p + ε t , t ∈ N , ● elementary properties ● the problem ● relation to literature ■ thinning operators θ 1 ◦ X t − 1 , . . . , θ p ◦ X t − p are independent Parametric models ■ Bernoulli variables in θ i ◦ X t − i survival-probability θ i Semiparametric model The unit root case Remarks: Summary ■ we follow definition of D U & L I (1991) (standard) ■ different definition than original by A L -O SH & A LZAID (1990) ■ THIS TALK: p = 1 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
Elementary properties: ■ first two (conditional) moments: Introduction ● definition INAR(1) ◆ E θ,G [ X t | X t − 1 ] = µ ε + θX t − 1 ● definition INAR( p ) ● elementary properties ◆ var θ,G [ X t | X t − 1 ] = σ 2 ε + θ (1 − θ ) X t − 1 ● the problem ● relation to literature ◆ same autocorrelation structure as AR( p ) 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: Introduction ● definition INAR(1) ◆ E θ,G [ X t | X t − 1 ] = µ ε + θX t − 1 ● definition INAR( p ) ● elementary properties ◆ var θ,G [ X t | X t − 1 ] = σ 2 ε + θ (1 − θ ) X t − 1 ● the problem ● relation to literature ◆ same autocorrelation structure as AR( p ) Parametric models ■ X is a Markov chain with transition probabilities Semiparametric model P θ,G The unit root case x t − 1 ,x t = (Binomial θ,x t − 1 ∗ G )( x t ) 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: Introduction ● definition INAR(1) ◆ E θ,G [ X t | X t − 1 ] = µ ε + θX t − 1 ● definition INAR( p ) ● elementary properties ◆ var θ,G [ X t | X t − 1 ] = σ 2 ε + θ (1 − θ ) X t − 1 ● the problem ● relation to literature ◆ same autocorrelation structure as AR( p ) Parametric models ■ X is a Markov chain with transition probabilities Semiparametric model P θ,G The unit root case x t − 1 ,x t = (Binomial θ,x t − 1 ∗ G )( x t ) Summary ■ stationary distribution, ν θ,G exists if 0 ≤ θ < 1 and µ G < ∞ ◆ for p = 1 well-known ◆ in general: no explicit formula for ν θ,G ◆ if E ε k 1 < ∞ then E ν θ,G X k 0 < ∞ for k = 1 , 2 , 3 ◆ existence facilitates asymptotic analysis 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
The problem: Given is Introduction ● definition INAR(1) ■ Parametric model: ● definition INAR( p ) ● elementary properties G known or belongs to smooth parametric model ● the problem ● relation to literature ■ Semiparametric model: Parametric models G unknown Semiparametric model The unit root case Goal: Summary given observations X 0 , . . . , X n estimate the parameters in the model efficiently 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
Relation to previous work: ■ estimation of θ : Introduction ● definition INAR(1) ◆ in parametric model: ● definition INAR( p ) ● elementary properties ■ F RANKE & S ELIGMANN (1993): ML (only p = 1 , ● the problem ● relation to literature limit-distribution derived but no efficiency proof) ■ B RÄNNÄS & H ALL (2001): GMM Parametric models ◆ in semiparametric model: Semiparametric model ■ D U & L I (1991): OLS The unit root case ■ S ILVA & O LIVEIRA (2005): spectral based Summary ■ estimation of G : ◆ even inefficient estimation of G not considered before 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
Parametric model: The model: Introduction ■ θ ∈ (0 , 1) Parametric models ● the model ■ G ∈ G A = ( G α | α ∈ A ) ● efficient estimation (1) ● efficient estimation (2) ◆ G α has finite third moment ● efficient estimation (3) ◆ A ⊂ R q open and convex Semiparametric model ◆ smoothness conditions on α �→ G α The unit root case Summary Goal: Given observations X 0 , . . . , X n estimate ( θ, α ) efficiently THIS TALK: G known 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
Model is Locally Asymptotically Normal: Model has LAN property, i.e. for u ∈ R : Introduction Parametric models d P ( n ) ● the model θ + u/ √ n uS n ( θ ) − u 2 � � ● efficient estimation (1) ( X 0 , . . . , X n ) = exp 2 I θ + o P ν,θ,G (1) , ● efficient estimation (2) d P ( n ) ● efficient estimation (3) θ Semiparametric model d The unit root case where the score S n ( θ ) → N(0 , I θ ) under P θ − 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 : Introduction Parametric models d P ( n ) ● the model θ + u/ √ n uS n ( θ ) − u 2 � � ● efficient estimation (1) ( X 0 , . . . , X n ) = exp 2 I θ + o P ν,θ,G (1) , ● efficient estimation (2) d P ( n ) ● efficient estimation (3) θ Semiparametric model d The unit root case where the score S n ( θ ) → N(0 , I θ ) under P θ − This makes life tractable: Summary (d / d θ ) log P θ,G X t − 1 ,X t = E θ [ ˙ s θ,X t − 1 ( θ ◦ X t − 1 ) | X t , X t − 1 ] , where ˙ s θ,X t − 1 ( · ) is score of Binomial( θ, X t − 1 ) 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 : Introduction Parametric models d P ( n ) ● the model θ + u/ √ n uS n ( θ ) − u 2 � � ● efficient estimation (1) ( X 0 , . . . , X n ) = exp 2 I θ + o P ν,θ,G (1) , ● efficient estimation (2) d P ( n ) ● efficient estimation (3) θ Semiparametric model d The unit root case where the score S n ( θ ) → N(0 , I θ ) under P θ − This makes life tractable: Summary (d / d θ ) log P θ,G X t − 1 ,X t = E θ [ ˙ s θ,X t − 1 ( θ ◦ X t − 1 ) | X t , X t − 1 ] , where ˙ s θ,X t − 1 ( · ) is score of Binomial( θ, X t − 1 ) Intuition: � additional observation θ ◦ X t − 1 transition-score equals s θ,X t − 1 ( θ ◦ X t − 1 ) ˙ � only observe X t − 1 , X t = ⇒ loss of information = ⇒ transition-score is coarsened 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
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