Periodic INAR(1) model with Bell innovations distribution

Abderrahmen Manaa

doi:10.1515/mcma-2024-2015

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Periodic INAR(1) model with Bell innovations distribution

Abderrahmen Manaa

Published/Copyright: September 26, 2024

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From the journal Monte Carlo Methods and Applications Volume 30 Issue 4

Abstract

In this paper, we introduce a class of periodic integer-valued autoregressive BL-PINAR(1) models with Bell innovations distribution based on the binomial thinning operator. The basic probabilistic and statistical properties of this class are studied. Indeed, the first and the second moment periodically stationary conditions are established. The closed forms of these moments are, under the obtained conditions, derived. Furthermore, the periodic autocovariance structure is also considered while providing the closed form of the periodic autocorrelation function. The conditional least squares (CLS), Yule–Walker (YW), weighted conditional least squares (WCLS), and conditional maximum likelihood (CML) methods are applied to estimate the underlying parameters. The asymptotic properties of the CLS and the YW estimators are obtained. The performances of these methods are compared through a simulation study. An application on a real data set is provided.

Keywords: Bell distribution; BL-PINAR model; periodic stationarity conditions

MSC 2020: 62F12; 62M10

A Appendix

Proof of Proposition 3.1

Substituting successively, 𝑚 times, in (2.2), we obtain

y t = ( ∏ i = 1 m φ t − i + 1 ) ∘ y t − m + ∑ j = 0 m − 1 ( ∏ i = 1 j φ t − i + 1 ) ∘ ε t − j , t ∈ Z .

Letting t = s + τ ⁢ S , s = 1 , … , S and τ ∈ Z , while putting m = S , and taking into account the periodicity of the parameter φ s , we obtain

y s + τ ⁢ S = ( ∏ i = 1 S φ i ) ∘ y s + ∑ j = 1 S ( ∏ i = 1 j − 1 φ s − i + 1 ) ∘ ε s − j + 1 + τ ⁢ S , t ∈ Z .

Then the mean μ y , s = E ⁢ ( y s + τ ⁢ S ) of the stochastic process { y s + τ ⁢ S ; τ ∈ Z } does not depend on 𝜏 and depends only on the season 𝑠, i.e., the process is periodically stationary in the mean, if and only if ∏ i = 1 S φ i < 1 ; hence

μ y , s = ( 1 − ∏ i = 1 S φ i ) − 1 ⁢ ∑ j = 1 S ( ∏ i = 1 j − 1 φ s − i + 1 ) ⁢ λ s − j + 1 ⁢ e λ s − j + 1 , s = 1 , … , S . ∎

Proof of Proposition 3.2

The variance γ ( t ) ⁢ ( 0 ) of the periodically correlated process y t = φ t ∘ y t − 1 + ε t , t ∈ Z , is given by the following non-homogeneous difference equation:

γ y ( t ) ⁢ ( 0 ) = φ t 2 ⁢ γ y ( t − 1 ) ⁢ ( 0 ) + ψ t ,

where ψ t = φ t ⁢ ( 1 − φ t ) ⁢ E ⁢ ( y t − 1 ) + λ t ⁢ ( 1 + λ t ) ⁢ e λ t . Iterating the last equation 𝑚 times, we obtain

γ y ( t ) ⁢ ( 0 ) = ( ∏ i = 1 m φ t − i + 1 2 ) ⁢ γ y ( t − m ) ⁢ ( 0 ) + ∑ j = 1 m ( ∏ i = 1 j − 1 φ t − i + 1 2 ) ⁢ ψ t − j + 1 .

Replacing, in the previous expression, 𝑡 by t = s + τ ⁢ S , s = 1 , … , S , τ ∈ Z , and then posing m = S , we obtain, while taking into account the periodicity of the coefficients,

γ ( s + τ ⁢ S ) ⁢ ( 0 ) = ( ∏ i = 1 S φ i 2 ) τ ⁢ γ ( s ) ⁢ ( 0 ) + ∑ k = 0 τ − 1 ( ∏ i = 1 S φ i 2 ) k ⁢ ∑ j = 1 S ( ∏ i = 1 j − 1 φ s − i + 1 2 ) ⁢ ψ s − j + 1 .

It is well known that the series ∑ k = 0 τ − 1 ( ∏ i = 1 S φ i 2 ) k converges if and only if the quantity ∏ i = 1 S φ i 2 < 1 (hence ∏ i = 1 S φ i < 1 ), and in this case, we have

γ y ( s ) ⁢ 0 = ( 1 − ∏ i = 1 S φ i 2 ) − 1 ⁢ ∑ j = 1 S ( ∏ i = 1 j − 1 φ s − i + 1 2 ) ⁢ ψ s − j + 1 ,

with ψ s = φ s ⁢ ( 1 − φ s ) ⁢ μ y , s − 1 + λ s ⁢ ( 1 + λ s ) ⁢ e λ s . ∎

Proof of Proposition 3.3

It is possible to deduce from [6, Proposition B] that

∏ i = 1 S φ i ⇔ η < 1 ,

which implies that all roots of the characteristic polynomial of 𝐴 are contained within the unit circle. Furthermore, if E ⁢ ∥ ζ τ ∥ < ∞ , then [9, Theorem 1] states that there exists a strict periodic stationary BL-PINAR(1) process satisfying (2.2). ∎

Proof of Proposition 4.1

The variances γ y ( s ) ⁢ ( 0 ) , s = 1 , … , S , were given in Proposition 3.2. Concerning the autocovariance function, we have γ y ( t ) ⁢ ( h ) = E ⁢ ( y t ⁢ y t − h ) − μ y , t ⁢ μ y , t − h , h ≥ 1 , which can be calculated as follows:

γ Y ( t ) ⁢ ( h ) = Cov ⁡ ( y t ; y t − h ) = Cov ⁡ ( φ t ∘ y t − 1 + ε t ; y t − h ) = Cov ⁡ ( φ t ∘ y t − 1 ; y t − h ) = φ t ⁢ γ y ( t − 1 ) ⁢ ( h − 1 ) , h ≥ 1 .

By iteration, we obtain

γ y ( t ) ⁢ ( h ) = ( ∏ i = 1 h φ t − i + 1 ) ⁢ γ y ( t − h ) ⁢ ( 0 ) , h ≥ 1 .

Letting t = s + τ ⁢ S , s = 1 , … , S , τ ∈ Z , and h = ν + k ⁢ S , ν = 1 , … , S and k ∈ N , then the preceding equality can be rewritten as follows:

γ y ( s ) ⁢ ( ν + k ⁢ S ) = ( ∏ i = 1 ν + k ⁢ S φ s − i + 1 ) ⁢ γ y ( s + τ ⁢ S − ( ν + k ⁢ S ) ) ⁢ ( 0 ) = ( ∏ i = 1 S φ i ) k ⁢ ( ∏ i = 1 ν φ s − i + 1 ) ⁢ γ y ( s − ν ) ⁢ ( 0 ) , h ≥ 1 ;

hence, we have

ρ ( s ) ⁢ ( ν + k ⁢ S ) = ( ∏ i = 1 ν φ s − i + 1 ) ⁢ ( ∏ i = 1 S φ i ) k ⁢ γ y ( s − ν ) ⁢ ( 0 ) / γ y ( s ) ⁢ ( 0 ) . ∎

Proof of Proposition 5.1

Using the well-known property

E ⁢ ( a ∘ X ) = a ⁢ E ⁢ ( X ) and E ⁢ [ ( a ∘ X ) 2 ] = a 2 ⁢ E ⁢ ( X 2 ) + a ⁢ ( 1 − a ) ⁢ E ⁢ ( X )

(for more properties, see [24]), according to Lemma 2.1, we can obtain the conditional mean

E ⁢ ( y t ∣ y t − 1 ) = φ t ⁢ y t − 1 + E ⁢ ( ε t ) = φ t ⁢ y t − 1 + μ ε , t .

Hence, to estimate the parameters ( φ s , μ ε , s ) , s = 1 , … , 𝑆, we consider a realization of a finite size 𝑁 below y ¯ s = ( y s , y s + S , … , y s + ( N − 1 ) ⁢ S ) , so we have the quadratic form

Q ⁢ ( φ t , μ ε , t ∣ y 1 , … , y n ) = ∑ t = 1 n [ y t − E ⁢ ( y t ∣ y t − 1 ) ] 2 = ∑ t = 1 n [ y t − ( φ t ⁢ y t − 1 + μ ε , t ) ] 2 .

Letting t = s + τ ⁢ S , s = 1 , … , S and τ ∈ Z , while assuming that n = N ⁢ S , and taking into account the periodicity of the parameters φ t and μ ε , t , we may rewrite the previous expression as follows:

Q ⁢ ( φ s , μ ε , s ∣ y ¯ s , y ¯ s − 1 ) = ∑ s = 1 S ∑ τ = 0 N − 1 [ y s + τ ⁢ S − ( φ s ⁢ y s − 1 + τ ⁢ S + μ ε , s ) ] 2 ;

hence, we have

(A.1) ∑ τ = 0 N − 1 [ y s + τ ⁢ S ⁢ y s − 1 + τ ⁢ S − ( φ s ⁢ y s − 1 + τ ⁢ S 2 + μ ε , s ⁢ y s − 1 + τ ⁢ S ) ] = 0 ,

(A.2) ∑ τ = 0 N − 1 [ y s + τ ⁢ S − ( φ s ⁢ y s − 1 + τ ⁢ S + μ ε , s ) ] = 0 .

From (A.1) and (A.2), we obtain

∑ τ = 0 N − 1 y s + τ ⁢ S ⁢ y s − 1 + τ ⁢ S − ( φ s ⁢ ∑ τ = 0 N − 1 y s − 1 + τ ⁢ S 2 + μ ε , s ⁢ ∑ τ = 0 N − 1 y s − 1 + τ ⁢ S ) = 0 , ∑ τ = 0 N − 1 y s + τ ⁢ S − ( φ s ⁢ ∑ τ = 0 N − 1 y s − 1 + τ ⁢ S + N ⁢ μ ε , s ) = 0 .

Hence, we have the system

φ s ⁢ ( ∑ τ = 0 N − 1 y s − 1 + τ ⁢ S 2 ) + μ ε , s ⁢ ( ∑ τ = 0 N − 1 y s − 1 + τ ⁢ S ) = ∑ τ = 0 N − 1 y s + τ ⁢ S ⁢ y s − 1 + τ ⁢ S , φ s ⁢ ( ∑ τ = 0 N − 1 y s − 1 + τ ⁢ S ) + N ⁢ μ ε , s = ∑ τ = 0 N − 1 y s + τ ⁢ S ,

which leads to

φ ̂ s = N ⁢ ∑ τ = 0 N − 1 [ ( y s + τ ⁢ S − 1 N ⁢ ∑ τ = 0 N − 1 y s + τ ⁢ S ) ⁢ ( y s − 1 + τ ⁢ S − 1 N ⁢ ∑ τ = 0 N − 1 y s + τ ⁢ S ) ] N ⁢ ∑ τ = 0 N − 1 ( y s − 1 + τ ⁢ S − 1 N ⁢ ∑ τ = 0 N − 1 y s − 1 + τ ⁢ S ) 2 , μ ̂ ε , s = ∑ τ = 0 N − 1 y s + τ ⁢ S − φ ̂ s ⁢ ∑ τ = 0 N − 1 y s − 1 + τ ⁢ S N , s = 1 , … , S .

The conditional mean of the CLS estimators φ ̂ s and μ ̂ ε , s , s = 1 , … , S , with respect to the 𝜎-field

F s = { y s , y s + S , … , y s + ( N − 1 ) ⁢ S } ,

are given by

E ⁢ ( φ ̂ s ∣ F s − 1 ) = ∑ τ = 0 N − 1 E ⁢ ( y s + τ ⁢ S ∣ y s − 1 + τ ⁢ S ) ⁢ y s − 1 + τ ⁢ S − ∑ τ = 0 N − 1 E ⁢ ( y s + τ ⁢ S ∣ y s − 1 + τ ⁢ S ) ⁢ y ̄ s − 1 ∑ τ = 0 N − 1 ( y s − 1 + τ ⁢ S − y ̄ s − 1 ) 2 ,

where E ⁢ ( y s + τ ⁢ S ∣ y s − 1 + τ ⁢ S ) = φ s ⁢ y s − 1 + τ ⁢ S + μ ε , s , s = 1 , … , S ; hence, we have, for s = 1 , … , S ,

E ⁢ ( φ ̂ s ∣ F s − 1 ) = ∑ τ = 0 N − 1 ( φ s ⁢ y s − 1 + τ ⁢ S + μ ε , s ) ⁢ y s − 1 + τ ⁢ S − ∑ τ = 0 N − 1 ( φ s ⁢ y s − 1 + τ ⁢ S + μ ε , s ) ⁢ y ̄ s − 1 ∑ τ = 0 N − 1 ( y s − 1 + τ ⁢ S − y ̄ s − 1 ) 2 .

Hence, we have E ⁢ ( φ ̂ s ∣ F s − 1 ) = φ s , s = 1 , … , S . The conditional mean for μ ̂ ε , s is

E ⁢ ( μ ̂ ε , s ∣ F s − 1 ) = N − 1 ⁢ ( ∑ τ = 0 N − 1 E ⁢ ( y s + τ ⁢ S ∣ y s − 1 + τ ⁢ S ) − N ⁢ φ s ⁢ y ̄ s − 1 ) ,

Hence, with simple manipulations, we obtain E ⁢ ( μ ̂ ε , s ∣ F s − 1 ) = μ ε , s , s = 1 , … , S . ∎

Proof of Proposition 5.2

It is easy check that ∂ g / ∂ θ i , s and ∂ 2 g / ∂ θ i , s ⁢ ∂ θ j , s for i , j ∈ { 1 , 2 } , s = 1 , … , S , with g ⁢ ( θ ¯ t , y t − 1 ) = E ⁢ ( y t ∣ y t − 1 ) , satisfy all the regularity conditions proposed in [15, p. 634]. Consequently, by [15, Theorem 3.1], we conclude that CLS-vector estimators θ ¯ ̂ s , CLS are strongly consistent. Then we have to prove that the following three conditions hold.

E ⁢ ( y t ∣ y t − 1 , y t − 2 , … , y 0 ) = E ⁢ ( y t ∣ y t − 1 ) , t ≥ 1 a.e.
E ⁢ [ U s + τ ⁢ S ⁢ ( θ ¯ s ) ⁢ | ( ∂ g ⁢ ( θ ¯ s , y s − 1 + τ ⁢ S ) / ∂ θ i , s ) ⁢ ( ∂ g ⁢ ( θ ¯ s , y s − 1 + τ ⁢ S ) / ∂ θ j , s ) | ] < ∞ , i = 1 , 2 , where
U s + τ ⁢ S ⁢ ( θ ¯ s ) = ( y s + τ ⁢ S − g ⁢ ( θ ¯ s , y s − 1 + τ ⁢ S ) ) 2 .
Σ s is non-singular.

Condition (A) is satisfied since y s + τ ⁢ S is a first-order Markov chain while 𝑠 is fixed in { 1 , … , S } . In order to prove condition (B), we check that the components of the matrix Ω s are all finite, i.e.,

( Ω s ) 1 , 1 = E ⁢ [ U s + τ ⁢ S 2 ⁢ ( θ ¯ s ) ⁢ ( ∂ ∂ φ s ⁢ g ⁢ ( θ ¯ s , y s − 1 + τ ⁢ S ) ) 2 ] = E ⁢ ( y s − 1 + τ ⁢ S 2 ⁢ U s + τ ⁢ S 2 ⁢ ( θ ¯ s ) ) = E ⁢ ( y s − 1 + τ ⁢ S 2 ⁢ E ⁢ ( U s + τ ⁢ S 2 ⁢ ( θ ¯ s ) ∣ y s − 1 + τ ⁢ S ) ) = E ⁢ ( y s − 1 + τ ⁢ S 2 ⁢ Var ⁡ ( y s + τ ⁢ S ∣ y s − 1 + τ ⁢ S ) ) = φ s ⁢ ( 1 − φ s ) ⁢ E ⁢ ( y s − 1 + τ ⁢ S 3 ) + σ ε , s 2 ⁢ E ⁢ ( y s − 1 + τ ⁢ S 2 ) = α 1 , s ⁢ ( 1 − α 1 , s ) ⁢ μ y , s − 1 ( 3 ) + σ ε , s 2 ⁢ μ y , s − 1 ( 2 ) < ∞ .

Similarly, we have

( Ω s ) 2 , 2 = E ⁢ [ U s + τ ⁢ S 2 ⁢ ( θ ¯ s ) ⁢ ( ∂ ∂ μ ε , s ⁢ g ⁢ ( θ ¯ s , y s − 1 + τ ⁢ S ) ) 2 ] = E ⁢ ( E ⁢ ( U s + τ ⁢ S 2 ⁢ ( θ ¯ s ) ∣ y s − 1 + τ ⁢ S ) ) = E ⁢ ( Var ⁡ ( y s + τ ⁢ S ∣ y s − 1 + τ ⁢ S ) ) = φ s ⁢ ( 1 − φ s ) ⁢ E ⁢ ( y s − 1 + τ ⁢ S ) + α 2 , s ⁢ ( 1 − α 2 , s ) ⁢ E ⁢ ( y s − 1 + τ ⁢ S ) + σ ε , s 2 = φ s ⁢ ( 1 − φ s ) ⁢ μ y , s − 1 + σ ε , s 2 < ∞ ,

( Ω s ) 1 , 2 = ( Ω s ) 2 , 1 = E ⁢ [ U s + τ ⁢ S 2 ⁢ ( θ ¯ s ) ⁢ ∂ ∂ φ s ⁢ g ⁢ ( θ ¯ s , y s − 1 + τ ⁢ S ) ⁢ ∂ ∂ μ ε , s ⁢ g ⁢ ( θ ¯ s , y s − 1 + τ ⁢ S ) ] = E ⁢ ( y s − 1 + τ ⁢ S ⁢ E ⁢ ( U s + τ ⁢ S 2 ⁢ ( θ ¯ s ) ∣ y s − 1 + τ ⁢ S ) ) = E ⁢ ( y s − 1 + τ ⁢ S ⁢ Var ⁡ ( y s + τ ⁢ S ∣ y s − 1 + τ ⁢ S ) ) = φ s ⁢ ( 1 − φ s ) ⁢ E ⁢ ( y s − 1 + τ ⁢ S 2 ) + σ ε , s 2 ⁢ E ⁢ ( y s − 1 + τ ⁢ S ) = φ s ⁢ ( 1 − φ s ) ⁢ μ y , s − 1 ( 2 ) + σ ε , s 2 ⁢ μ y , s − 1 < ∞ .

Finally, the matrix Ω s is given as follows:

Ω s = ( φ s ⁢ ( 1 − φ s ) ⁢ μ y , s − 1 ( 3 ) + σ ε , s 2 ⁢ μ y , s − 1 ( 2 ) φ s ⁢ ( 1 − φ s ) ⁢ μ y , s − 1 ( 2 ) + σ ε , s 2 ⁢ μ y , s − 1 φ s ⁢ ( 1 − φ s ) ⁢ μ y , s − 1 ( 2 ) + σ ε , s 2 ⁢ μ y , s − 1 φ s ⁢ ( 1 − φ s ) ⁢ μ y , s − 1 + σ ε , s 2 )

Therefore, condition (B) is also satisfied; then the elements of the matrix Σ s are given by

( Σ s ) 1 , 1 = E ⁢ [ ( ∂ ∂ φ s ⁢ g ⁢ ( θ ¯ s , y s − 1 + τ ⁢ S ) ) 2 ] = E ⁢ ( y s − 1 + τ ⁢ S 2 ) = μ y , s − 1 ( 2 ) , ( Σ s ) 2 , 2 = E ⁢ [ ( ∂ ∂ μ ε , s ⁢ g ⁢ ( θ ¯ s , y s − 1 + τ ⁢ S ) ) 2 ] = 1 , ( Σ s ) 1 , 2 = ( Σ s ) 2 , 1 = E ⁢ [ ∂ ∂ φ s ⁢ g ⁢ ( θ ¯ s , y s − 1 + τ ⁢ S ) ⁢ ∂ ∂ μ ε , s ⁢ g ⁢ ( θ ¯ s , y s − 1 + τ ⁢ S ) ] = E ⁢ ( y s − 1 + τ ⁢ S ) = μ y , s − 1 .

Hence, we have

Σ s = E ⁢ ( y s − 1 + τ ⁢ S 2 y s − 1 + τ ⁢ S y s − 1 + τ ⁢ S 1 ) = ( μ y , s − 1 ( 2 ) μ y , s − 1 μ y , s − 1 1 ) .

Then we have

Σ s − 1 = 1 μ y , s − 1 ( 2 ) − ( μ y , s − 1 ) 2 ⁢ ( 1 − μ y , s − 1 − μ y , s − 1 μ y , s − 1 ( 2 ) ) .

The calculation of the matrix’s ( Σ s ) determinant gives

| Σ s | = μ y , s − 1 ( 2 ) − ( μ y , s − 1 ) 2 = σ y , s − 1 2 > 0 ,

which leads us to conclude that the matrix Σ s is invertible. Thus, condition (C) is also satisfied. Finally, by Klimko and Nelson [15, Theorem 3.2], the CLS-vector estimators θ ¯ ̂ s , CLS are asymptotically normally distributed. This achieves the proof. ∎

Proof of Proposition 5.3

The proof is straightforwardly from Proposition 4.1. ∎

Proof of Proposition 5.4

This follows from straightforward modifications of the proof of [11, Theorem 3]. ∎

Acknowledgements

The author wishes to express gratitude to Professor Karl Sabelfeld for his invaluable support and patience.

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Received: 2024-04-04

Revised: 2024-09-09

Accepted: 2024-09-10

Published Online: 2024-09-26

Published in Print: 2024-12-01

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https://doi.org/10.1515/mcma-2024-2015

Keywords for this article

Bell distribution; BL-PINAR model; periodic stationarity conditions