Poisson INAR processes with serial and seasonal correlation Mrton - PowerPoint PPT Presentation

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)) X t − 1 � t ∈ Z , X t = ξ t , j + ε t , j = 1 { ξ t , j , : t ∈ Z , j ∈ N } and { ε t : t ∈ Z } are independent, non-negative, integer-valued, identically distributed r.v.’s P ( ξ 1 , 1 ∈ { 0 , 1 } ) = 1, i.e., ξ 1 , 1 has Bernoulli distribution Parameters: α := E ξ 1 , 1 , λ := E ε 1 , b 2 := Var ε 1 Reformulation: X t = α ◦ X t − 1 + ε t α < 1 α = 1 Classification: stable unstable Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Branching process with immigration (BPI) X k − 1 1 2 ✪ ☎ ❏ ✪ ☎ ❏ � ☞ ❏ � ☞ ✪ ☎ ❏ ✪ ☎ ❏ ❏ � ☞ ✪ ☎ ❏ ✪ ☎ ❏ � ☞ ❏ . . . 1 . . . 1 . . . 1 . . . 1 2 . . . ξ k , 1 ξ k , 2 ξ k , X k − 1 ε k � �� � � �� � immigration offsprings X k − 1 � X 0 = 0 X k = ξ k , j + ε k , j = 1 { ξ k , j , ε k : j ∈ N , k ∈ Z + } independent { ξ k , j : j ∈ N , k ∈ Z + } identically distributed { ε k : k ∈ Z + } identically distributed with P ( ε 1 � = 0 ) > 0 Parameters: m := E ξ 1 , 1 , σ 2 = Var ξ 1 , 1 , λ := E ε 1 , b 2 := Var ε 1 m > 1 m < 1 m = 1 Classification: supercritical subcritical critical Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Conditional structure Filtration: F k := σ ( X 0 , X 1 , . . . , X k ) , k ∈ Z + Conditional expectation: E ( X k | F k − 1 ) = mX k − 1 + λ M k := X k − E ( X k | F k − 1 ) = X k − mX k − 1 − λ , k ∈ N martingale differences, and we have X k = λ + mX k − 1 + M k Conditional variance: E ( M 2 k | F k − 1 ) = σ 2 X k − 1 + b 2 since X k − 1 � M k = X k − mX k − 1 − λ = ( ξ k , j − m ) + ( ε k − λ ) j = 1 Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Autoregressive process (AR) AR(1) model t ∈ Z X t = µ + α X t − 1 + ε t , µ ∈ R is the drift, α ∈ R is the autoregressive parameter, and { ε t , t ∈ Z } is a sequence of martingale differences α > 1 α < 1 α = 1 Classification: explosive stable unstable 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)) Y t − s � t ∈ Z , Y t = ξ t , j + ε t , j = 1 { ξ t , j : t ∈ Z , j ∈ N } and { ε t : t ∈ Z } are independent, non-negative, integer-valued, identically distributed r.v.’s P ( ξ 1 , 1 ∈ { 0 , 1 } ) = 1, i.e., ξ 1 , 1 has Bernoulli distribution s ∈ N denotes the seasonal period Parameters: φ := E ξ 1 , 1 , λ := E ε 1 Reformulation: Y t = φ ◦ Y t − s + ε t φ < 1 φ = 1 Classification: stable 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 { ε t : t ∈ Z } as ε t − sk � � � ∞ ∞ φ k ◦ ε t − ks = ε t + d Y t = Z t , k , j , t ∈ Z , k = 0 k = 1 j = 1 where d = stands for equality in distribution and Z t , k , j ∼ B e ( φ k ) . Let { ε t : 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 Y t ∼ P o ( λ/ ( 1 − φ )) and the autocorrelation function is given by � φ k / s , if k is a multiple of s , ρ ( k ) = 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. 1.0 5 0.8 4 0.6 3 ACF 0.4 y t 2 0.2 1 0.0 −0.2 0 0 20 40 60 80 100 0 10 20 30 40 50 Time Lag 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 n � [ Y t − E θ ( Y t |F t − 1 )] 2 � θ CLS := arg min θ t = s + 1 with E θ ( Y t |F t − 1 ) = E θ ( Y t | Y t − s ) = g ( θ, Y t − s ) , where g ( θ, y ) := φ y + λ . Solving the normal equations we have n n n � � � ( n − s ) Yt Yt − s − Yt Yt − s � t = s + 1 t = s + 1 t = s + 1 φ CLS := � � 2 n n � � Y 2 ( n − s ) Yt − s t − s − t = s + 1 t = s + 1 � � � n � n � λ CLS := 1 Y t − � φ CLS Y t − s n − s t = s + 1 t = s + 1 Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

Asymptotic result for conditional least squares �� � �� 0 � � √ φ CLS − φ d n → N , Σ � 0 λ CLS − λ where � � λ − 1 φ ( 1 − φ ) 2 + ( 1 − φ 2 ) − ( 1 + φ ) λ Σ := λ + ( 1 + φ )( 1 − φ ) − 1 λ 2 − ( 1 + φ ) λ 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 n � ℓ ( θ ) = log P θ ( Y n , . . . , Y s | Y s − 1 , . . . , Y 0 ) = log [ P θ ( Y t | Y t − s )] , where t = s P θ ( Y t | Y t − s ) = [ B i ( Y t − s , φ ) ∗ P o ( λ )] ( Y t ) = e − λ � min ( Yt , Yt − s ) λ Yt − i Yt − s ( Yt − i )! ( i ) φ i ( 1 − φ ) Yt − s − i i = 0 Asymptotic result: �� � √ φ CML − φ → N ( 0 , I − 1 ( θ )) , d n � λ CML − λ 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 ) YW CLS CML YW CLS CML n φ 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. 20 Claims count 15 10 5 0 20 40 60 80 100 120 Time 0.4 0.4 0.2 0.2 PACF ACF 0.0 0.0 −0.2 −0.2 5 10 15 20 5 10 15 20 Lag Lag Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation