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Márton Ispány

S³ Seminar
January 30, 2015

Márton Ispány

(University of Debrecen, Hungary)

https://s3-seminar.github.io/seminars/marton-ispany

Title — Poisson INAR processes with serial and seasonal correlation

Abstract — Recently, there has been considerable interest in integer-valued time series models. Motivation to include discrete data models comes from the need to account for the discrete nature of certain data sets, often counts of events, objects or individuals. Among the most successful integer-valued time series models proposed in the literature we mention the INteger-valued AutoRegressive model of order p (INAR(p)). However, seasonal count processes have not been investigated yet, except one of our new papers. In the talk, we study INAR processes which possess serial and seasonal structure as well. The main properties of the models will be derived such as the stationarity and the autocorrelation function. The conditional least squares and conditional maximum likelihood estimators of the model parameters will be studied and their asymptotical properties will be established. In addition, we would like to discuss the case in which the marginal distributions are Poisson in detail. Monte Carlo experiments will be conducted to evaluate and compare the performance of various estimators for finite sample sizes. Real data set on the area of insurance will be applied to evaluate the model performance.

Biography — Márton Ispány received the M.Sc.(1989) and PhD (summa cum laude) in Statistics (1997) from University of Debrecen. Since 2007 he has been with the Department of Information Technology, Faculty of Informatics, University of Debrecen. Since 2012 he has been the head of the department. Márton Ispány 's recent research interests are in branching processes (functional limit theorems, asymptotics for conditional least squares estimation, integer valued autoregression), statistical modelling(generalized SVD, contaminated statistical models, EM algorithm), data mining (decision trees, stochastic algorithms, MCMC, web mining), and applied statistics: econometrics and insurance, cross-country modelling, statistical genetics.

S³ Seminar

January 30, 2015
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  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-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

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  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
    Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation

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  3. 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

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  4. 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

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  5. 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

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  6. 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

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  7. 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

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  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 {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

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  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.
    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

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  10. 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

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  11. 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

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  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
    (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

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  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
    0.30 −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

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  14. 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

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  15. 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

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  16. 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

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  17. 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

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  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 =

    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

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  19. 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

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  20. 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

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  21. 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

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  22. 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

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  23. 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

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  24. 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

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