Poisson INAR processes with serial and seasonal correlation Márton Ispány University of Debrecen, Faculty of Informatics Joint result with Marcelo Bourguignon, Klaus L. P. Vasconcellos, and Valdério A. Reisen Workshop on Time series and counting processes with application to environmental and networking problems Supélec January 30, 2015 The research was supported by the TÁMOP-4.2.2.C-11/1/KONV-2012-0001 project. The project has been supported by the European Union, co-financed by the European Social Fund. Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Outline • Integer valued autoregression and INAR(1) model • Comparison of AR, INAR, and branching processes • The purely seasonal INAR(1) model • Estimation methods • Simulation and real data examples • INAR process with serial and seasonal correlation • Stationarity and second order properties • Estimation methods Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Integer valued autoregression (INAR) INAR(1) model (Al-Osh and Alzaid (1987)) Xt = Xt−1 j=1 ξt,j + εt , t ∈ Z, {ξt,j, : t ∈ Z, j ∈ N} and {εt : t ∈ Z} are independent, non-negative, integer-valued, identically distributed r.v.’s P(ξ1,1 ∈ {0, 1}) = 1, i.e., ξ1,1 has Bernoulli distribution Parameters: α := E ξ1,1 , λ := E ε1 , b2 := Var ε1 Reformulation: Xt = α ◦ Xt−1 + εt Classification: α < 1 stable α = 1 unstable Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Branching process with immigration (BPI) 1 . . . ξk,1 7 7 7 ¤ ¤ ¤ t t t 1 1 . . . ξk,2 7 7 7 ¤ ¤ ¤ t t t 2 . . . 1 . . . ξk,Xk−1 t t t Xk−1 offsprings 1 2 . . . εk immigration Xk = Xk−1 j=1 ξk,j + εk , X0 = 0 {ξk,j, εk : j ∈ N, k ∈ Z+} independent {ξk,j : j ∈ N, k ∈ Z+} identically distributed {εk : k ∈ Z+} identically distributed with P(ε1 = 0) > 0 Parameters: m := E ξ1,1 , σ2 = Var ξ1,1 , λ := E ε1 , b2 := Var ε1 Classification: m < 1 subcritical m = 1 critical m > 1 supercritical Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Conditional structure Filtration: Fk := σ(X0, X1, . . . , Xk ), k ∈ Z+ Conditional expectation: E(Xk | Fk−1) = mXk−1 + λ Mk := Xk − E(Xk | Fk−1) = Xk − mXk−1 − λ, k ∈ N martingale differences, and we have Xk = λ + mXk−1 + Mk Conditional variance: E(M2 k | Fk−1) = σ2Xk−1 + b2 since Mk = Xk − mXk−1 − λ = Xk−1 j=1 (ξk,j − m) + (εk − λ) Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Autoregressive process (AR) AR(1) model Xt = µ + αXt−1 + εt , t ∈ Z µ ∈ R is the drift, α ∈ R is the autoregressive parameter, and {εt , t ∈ Z} is a sequence of martingale differences Classification: α < 1 stable α = 1 unstable α > 1 explosive Connection • All INAR(1) process is a branching process with immigration. • All branching process with immigration is an AR(1) processes with drift and conditionally heteroscedasticity. Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

INAR(1) process with a seasonal structure INAR(1)s model (Bourguignon, Vasconcellos, Reisen, I (2014)) Yt = Yt−s j=1 ξt,j + εt , t ∈ Z, {ξt,j : t ∈ Z, j ∈ N} and {εt : t ∈ Z} are independent, non-negative, integer-valued, identically distributed r.v.’s P(ξ1,1 ∈ {0, 1}) = 1, i.e., ξ1,1 has Bernoulli distribution s ∈ N denotes the seasonal period Parameters: φ := E ξ1,1 , λ := E ε1 Reformulation: Yt = φ ◦ Yt−s + εt Classification: φ < 1 stable φ = 1 unstable Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Stationarity and second order properties If φ ∈ [0, 1), the unique stationary marginal distribution of INAR(1)s model can be expressed in terms of {εt : t ∈ Z} as Yt d = ∞ k=0 φk ◦ εt−ks = εt + ∞ k=1 εt−sk j=1 Zt,k,j, t ∈ Z, where d = stands for equality in distribution and Zt,k,j ∼ Be(φk ). Let {εt : t ∈ Z} be an i.i.d. sequence of Poisson distributed variables with mean λ ∈ R+ and let φ ∈ [0, 1). Then the unique stationary solution satisfies Yt ∼ Po(λ/(1 − φ)) and the autocorrelation function is given by ρ(k) = φk/s, if k is a multiple of s, 0, otherwise. Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Sample path and its sample ACF 100 simulated values of the INAR(1)s process and its sample autocorrelation function for φ = 0.5, λ = 1 and s = 12. Time yt 0 20 40 60 80 100 0 1 2 3 4 5 0 10 20 30 40 50 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 Lag ACF Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Estimation methods: conditional least squares (CLS) The conditional least squares estimator of θ = (φ, λ)T is given by θCLS := arg min θ n t=s+1 [Yt − Eθ(Yt |Ft−1)]2 with Eθ(Yt |Ft−1) = Eθ(Yt |Yt−s) = g(θ, Yt−s), where g(θ, y) := φy + λ. Solving the normal equations we have φCLS := (n−s) n t=s+1 Yt Yt−s− n t=s+1 Yt n t=s+1 Yt−s (n−s) n t=s+1 Y2 t−s− n t=s+1 Yt−s 2 λCLS := 1 n−s n t=s+1 Yt −φCLS n t=s+1 Yt−s Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Asymptotic result for conditional least squares √ n φCLS − φ λCLS − λ d → N 0 0 , Σ where Σ := λ−1φ(1 − φ)2 + (1 − φ2) −(1 + φ)λ −(1 + φ)λ λ + (1 + φ)(1 − φ)−1λ2 Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Estimation methods: conditional maximum likelihood (CML) The INAR(1)s process consists of s mutually independent INAR(1) processes, thus it is an s-step Markov chain. Hence, the conditional log-likelihood function is given by (θ) = log Pθ(Yn, . . . , Ys|Ys−1, . . . , Y0) = n t=s log[Pθ(Yt |Yt−s)], where Pθ(Yt |Yt−s) = [Bi(Yt−s, φ) ∗ Po(λ)] (Yt ) =e−λ min(Yt ,Yt−s) i=0 λYt −i (Yt −i)! (Yt−s i )φi (1−φ)Yt−s−i Asymptotic result: √ n φCML − φ λCML − λ d → N(0, I−1(θ)), where I(θ) is a 2 × 2 Fisher information matrix. Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Monte Carlo simulation study Table: Biases of estimators for λ = 1 (MSE in parenthesis) Bias(φ)/MSE(φ) Bias(λ)/MSE(λ) n φ YW CLS CML YW CLS CML 0.30 −0.0178 −0.0307 −0.0067 0.0315 0.0508 0.0040 (0.0125) (0.0133) (0.0114) (0.0353) (0.0365) (0.0291) 100 0.50 −0.0240 −0.0334 −0.0081 0.0500 0.0691 0.0064 (0.0100) (0.0116) (0.0063) (0.0549) (0.0560) (0.0304) 0.80 −0.0267 −0.0362 −0.0031 0.1385 0.1854 0.0113 (0.0058) (0.0078) (0.0012) (0.1583) (0.1921) (0.0289) 0.30 −0.0115 −0.0156 −0.0067 0.0221 0.0282 0.0115 (0.0045) (0.0044) (0.0035) (0.0130) (0.0133) (0.0104) 250 0.50 −0.0106 −0.0146 −0.0029 0.0254 0.0337 0.0057 (0.0037) (0.0040) (0.0023) (0.0174) (0.0184) (0.0109) 0.80 −0.0143 −0.0166 −0.0016 0.0700 0.0823 0.0028 (0.0019) (0.0022) (0.0004) (0.0511) (0.0572) (0.0113) 0.30 −0.0058 −0.0079 −0.0023 0.0063 0.0093 −0.0008 (0.0022) (0.0022) (0.0018) (0.0055) (0.0056) (0.0045) 500 0.50 −0.0033 −0.0056 −0.0007 0.0102 0.0148 0.0029 (0.0018) (0.0018) (0.0010) (0.0087) (0.0089) (0.0052) 0.80 −0.0086 −0.0098 −0.0003 0.0468 0.0500 0.0043 (0.0009) (0.0009) (0.0002) (0.0237) (0.0255) (0.0055) Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Real data example (Freeland) Monthly counts of claims of short-term disability benefits reported to the Richmond, BC Workers Compensation Board. Time Claims count 0 20 40 60 80 100 120 5 10 15 20 5 10 15 20 −0.2 0.0 0.2 0.4 Lag ACF 5 10 15 20 −0.2 0.0 0.2 0.4 Lag PACF Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Fitted models Model CML estimates CLS estimates AIC BIC INAR(1)12 φ 0.1746 (0.0036) 0.2410 (0.0899) 530.613 536.013 λ 5.1391 (0.1951) 4.7554 (0.5897) INAR(1) φ 0.4418 (0.0029) 0.5510 (0.0783) 538.469 543.869 λ 3.5224 (0.1364) 2.8526 (0.5079) The model fitted by CML estimation is Yt = 0.1746 ◦ Yt−12 + t , t ∼ Po(5.1391) Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

INAR(1) process with serial and seasonal structure Seasonal INAR({1,s}) model (I and Reisen (2014)) Zt = Zt−1 j=1 ξt,j + Zt−s j=1 ηt,j + εt , t ∈ Z, {ξt,j : t ∈ Z, j ∈ N}, {ηt,j : t ∈ Z, j ∈ N} and {εt : t ∈ Z} are independent, non-negative, integer-valued, i.d. r.v.’s ξ1,1 and η1,1 have Bernoulli distribution s ∈ N denotes the seasonal period Parameters: α := E ξ1,1 , φ := E η1,1 , λ := E ε1 Reformulation: Zt = α ◦ Zt−1 + φ ◦ Zt−s + εt Classification: α + φ < 1 stable α + φ = 1 unstable α + φ > 1 explosive Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

State space representation Zt = A ◦ Zt−1 + εt where A :=       α 0 · · · 0 φ 1 0 · · · 0 0 ... 0 · · · 1 0       Zt :=       Zt Zt−1 . . . Zt−s+1       εt :=       εt 0 . . . 0       The characteristic polynomial of A is given by det(xI − A) = xsP(x−1) where P denotes the autoregressive polynomial defined by P(x) := 1 − αx − φxs The INAR({1,s}) model is called primitive if the matrix A is primitive which holds iff α > 0 and φ > 0 Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Stationarity Lemma The roots of a primitive autoregressive polynomial P lie outside of the complex unit circle iff α + φ < 1. Then, for |x| 1, P(x)−1 = ∞ j=0 γj xj with ∞ j=0 γj < ∞. The non-negative sequence {γj : j ∈ Z+} satisfies the recursion γ0 = 1, γj = αγj−1 , j = 1, . . . , s − 1, γj = αγj−1 + φγj−s , j ≥ s. If α + φ < 1, the unique stationary marginal distribution of INAR({1,s}) model can be expressed in terms of {εt : t ∈ Z} as Zt d = ∞ k=0 γk ◦ εt−ks = εt + ∞ k=1 εt−sk j=1 Ut,k,j, Ut,k,j ∼ Be(γk ) Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Second order properties Let {εt : t ∈ Z} be an i.i.d. sequence of Poisson distributed variables with mean λ ∈ R+ and let φ ∈ [0, 1). Then the unique stationary solution satisfies Yt ∼ Po(λ/(1 − α − φ)). The autocorrelation function satisfies the recursion ρ(k) = αρ(k − 1) + φρ(k − s), k ∈ Z Recursive computation of the autocorrelation function starting from initial values ρ(0) = 1 and ρ(k) = αρ(k − 1) + φρ(s − k), k = 1, . . . , s − 1 The partial autocorrelation function satisfies τ(k) = 0, if k = 0, 1, . . . , s = 0, otherwise. Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Sample ACF and PACF (α = 0.3, φ = 0.5 and s = 12) 0 10 20 30 40 50 60 70 80 90 100 −0.2 0 0.2 0.4 0.6 0.8 Lag Sample Autocorrelation Sample Autocorrelation Function (ACF) 0 10 20 30 40 50 60 70 80 90 100 −0.2 0 0.2 0.4 0.6 0.8 Lag Sample Partial Autocorrelations Sample Partial Autocorrelation Function Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Estimation methods: conditional least squares (CLS) The conditional least squares estimator of θ = (α, φ, λ)T is given by θCLS := arg min θ n t=s+1 [Yt − Eθ(Yt |Ft−1)]2 with Eθ(Yt |Ft−1) = Eθ(Yt |Yt−1, Yt−s) = αYt−1 + φYt−s + λ. The normal equations are given by n t=s+1    Yt−1 Yt−s 1    Yt−1 Yt−s 1    α φ λ    = n t=s+1 Yt    Yt−1 Yt−s 1    Asymptotic result: √ n(θCLS − θ) d → N(0, Σ) Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Estimation methods: conditional maximum likelihood (CML) The conditional log-likelihood function is given by (θ) = log Pθ(Yn, . . . , Ys|Ys−1, . . . , Y0) = n t=s log[Pθ(Yt |Yt−1, Yt−s)], where Pθ(Yt |Yt−1, Yt−s) = [Bi(Yt−1, α) ∗ Bi(Yt−s, φ) ∗ Po(λ)] (Yt ) Asymptotic result: √ n    αCML − α φCML − φ λCML − λ    d → N(0, I−1(θ)), where I(θ) is a 3 × 3 Fisher information matrix. Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Real data example revisited The CLS estimates of parameters by solving the normal equations are α = 0.5388 φ = 0.1561 λ = 1.8011 Thank you! Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

References AL-OSH, M.A., ALZAID, A.A.: First-order integer valued autoregressive (INAR(1)) process. J. Time Ser. Anal. 8 (1987) 261–275. BARCZY, M. ISPÁNY, M., PAP, G.: Asymptotic behavior of unstable INAR(p) processes Stoch. Proc. Appl. 121 (2011) 583–608. BOURGUIGNON, B., ISPÁNY, M., REISEN, V., VASCONCELLOS: A Poisson INAR(1) process with a seasonal structure. J. Stat. Comp. Simul. accepted DU, J., LI, Y.: The integer-valued autoregressive (INAR(p)) model. J. Time Ser. Anal. 12 (1991) 129–142. Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation