Poisson INAR processes with serial and seasonal correlation

Transcription

1 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 C-11/1/KONV project. The project has been supported by the European Union, co-financed by the European Social Fund.

2 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

3 Integer valued autoregression (INAR) INAR(1) model (Al-Osh and Alzaid (1987)) X t 1 X t = ξ t,j + ε t, t Z, 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 Classification: α < 1 stable α = 1 unstable

4 Branching process with immigration (BPI) X k 1 1 }... ξ k, ξ k,2 {{ 1... ξ k,xk 1 } offsprings ε k }{{} immigration X k = X k 1 j=1 ξ k,j + ε k, X 0 = 0 {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 Classification: m < 1 subcritical m = 1 critical m > 1 supercritical

5 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(Mk 2 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

6 Autoregressive process (AR) AR(1) model X t = µ + αx t 1 + ε t, t Z µ R is the drift, α R is the autoregressive parameter, and {ε t, 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.

7 INAR(1) process with a seasonal structure INAR(1) s model (Bourguignon, Vasconcellos, Reisen, I (2014)) Y t s Y t = ξ t,j + ε t, t Z, 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

8 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 d Y t = φ k ε t ks = ε t + k=0 k=1 ε t sk j=1 Z t,k,j, t Z, where d = stands for equality in distribution and Z t,k,j Be(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 Po(λ/(1 φ)) and the autocorrelation function is given by { φ k/s, if k is a multiple of s, ρ(k) = 0, otherwise.

9 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. y t ACF Time Lag

10 Estimation methods: conditional least squares (CLS) The conditional least squares estimator of θ = (φ, λ) T is given by θ CLS := arg min θ n t=s+1 [Y t E θ (Y t F t 1 )] 2 with E θ (Y t F t 1 ) = E θ (Y t Y t s ) = g(θ, Y t s ), where g(θ, y) := φy + λ. Solving the normal equations we have φ CLS := (n s) λ CLS := 1 n s n Y t Y t s n n Y t Y t s t=s+1 t=s+1 t=s+1 ( ) n n 2 (n s) Y t s 2 Y t s t=s+1 t=s+1 ( ) n n Y t φ CLS Y t s t=s+1 t=s+1

11 Asymptotic result for conditional least squares where Σ := ) ( φcls φ n λ CLS λ d N [ λ 1 φ(1 φ) 2 + (1 φ 2 ) (( ) ) 0, Σ 0 (1 + φ)λ (1 + φ)λ λ + (1 + φ)(1 φ) 1 λ 2 ]

12 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 ) = [Bi(Y t s, φ) Po(λ)] (Y t ) Asymptotic result: =e λ min(y t,y t s ) i=0 n ( φcml φ λ CML λ ) λ Y t i (Y t i)!( Y t s)φ i (1 φ) Y t s i i d N (0, I 1 (θ)), where I(θ) is a 2 2 Fisher information matrix.

13 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 (0.0125) (0.0133) (0.0114) (0.0353) (0.0365) (0.0291) (0.0100) (0.0116) (0.0063) (0.0549) (0.0560) (0.0304) (0.0058) (0.0078) (0.0012) (0.1583) (0.1921) (0.0289) (0.0045) (0.0044) (0.0035) (0.0130) (0.0133) (0.0104) (0.0037) (0.0040) (0.0023) (0.0174) (0.0184) (0.0109) (0.0019) (0.0022) (0.0004) (0.0511) (0.0572) (0.0113) (0.0022) (0.0022) (0.0018) (0.0055) (0.0056) (0.0045) (0.0018) (0.0018) (0.0010) (0.0087) (0.0089) (0.0052) (0.0009) (0.0009) (0.0002) (0.0237) (0.0255) (0.0055)

14 Real data example (Freeland) Monthly counts of claims of short-term disability benefits reported to the Richmond, BC Workers Compensation Board. Claims count Time ACF PACF Lag Lag

15 Fitted models Model CML estimates CLS estimates AIC BIC INAR(1) 12 φ (0.0036) (0.0899) λ (0.1951) (0.5897) INAR(1) φ (0.0029) (0.0783) λ (0.1364) (0.5079) The model fitted by CML estimation is Y t = Y t 12 + ɛ t, ɛ t Po(5.1391)

16 INAR(1) process with serial and seasonal structure Seasonal INAR({1,s}) model (I and Reisen (2014)) Z t 1 Z t = j=1 Z t s ξ t,j + j=1 η t,j + ε t, t Z, {ξ t,j : t Z, j N}, {η t,j : t Z, j N} and {ε t : t Z} are independent, non-negative, integer-valued, i.d. r.v. s ξ 1,1 and η 1,1 have Bernoulli distribution s N denotes the seasonal period Parameters: α := E ξ 1,1, φ := E η 1,1, λ := E ε 1 Reformulation: Classification: Z t = α Z t 1 + φ Z t s + ε t α + φ < 1 α + φ = 1 stable unstable α + φ > 1 explosive

17 State space representation Z t = A Z t 1 + ε t where α 0 0 φ A :=... Z t := Z t Z t 1. Z t s+1 ε t ε 0 t :=. The characteristic polynomial of A is given by det(xi A) = x s P(x 1 ) where P denotes the autoregressive polynomial defined by P(x) := 1 αx φx s The INAR({1,s}) model is called primitive if the matrix A is primitive which holds iff α > 0 and φ > 0 0

18 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 = γ j x j j=0 with γ j <. j=0 The non-negative sequence {γ j : j Z + } satisfies the recursion γ 0 = 1, γ j = αγ j 1, j = 1,..., s 1, γ j = αγ j 1 + φγ j s, j s. If α + φ < 1, the unique stationary marginal distribution of INAR({1,s}) model can be expressed in terms of {ε t : t Z} as d ε t sk Z t = γ k ε t ks = ε t + U t,k,j, U t,k,j Be(γ k ) k=0 k=1 j=1

19 Second order properties 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 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 { 0, if k = 0, 1,..., s τ(k) = 0, otherwise.

20 Sample ACF and PACF (α = 0.3, φ = 0.5 and s = 12) Sample Autocorrelation Function (ACF) Sample Partial Autocorrelation Function Sample Autocorrelation Sample Partial Autocorrelations Lag Lag

21 Estimation methods: conditional least squares (CLS) The conditional least squares estimator of θ = (α, φ, λ) T is given by θ CLS := arg min θ n t=s+1 [Y t E θ (Y t F t 1 )] 2 with E θ (Y t F t 1 ) = E θ (Y t Y t 1, Y t s ) = αy t 1 + φy t s + λ. The normal equations are given by n Y t 1 [ ] α Y t s Y t 1 Y t s 1 φ = t=s+1 1 λ Asymptotic result: n( θcls θ) d N (0, Σ) n t=s+1 Y t Y t 1 Y t s 1

22 Estimation methods: conditional maximum likelihood (CML) The conditional log-likelihood function is given by l(θ) = log P θ (Y n,..., Y s Y s 1,..., Y 0 ) = where n log[p θ (Y t Y t 1, Y t s )], t=s P θ (Y t Y t 1, Y t s ) = [Bi(Y t 1, α) Bi(Y t s, φ) Po(λ)] (Y t ) Asymptotic result: α CML α n φ CML φ d N (0, I 1 (θ)), λ CML λ where I(θ) is a 3 3 Fisher information matrix.

23 Real data example revisited The CLS estimates of parameters by solving the normal equations are α = φ = λ = Thank you!

24 References AL-OSH, M.A., ALZAID, A.A.: First-order integer valued autoregressive (INAR(1)) process. J. Time Ser. Anal. 8 (1987) BARCZY, M. ISPÁNY, M., PAP, G.: Asymptotic behavior of unstable INAR(p) processes Stoch. Proc. Appl. 121 (2011) 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)