Asymptotic distribution of the Yule-Walker estimator for INAR(p) processes

Transcription

1 Asymptotic distribution of the Yule-Walker estimator for IAR(p) processes Isabel Silva a,b, M Eduarda Silva b,c a Departamento de Engenharia Civil, Faculdade de Engenharia - Universidade do Porto, Rua Dr Roberto Frias, Porto, PORTUGAL b Departamento de Matemática Aplicada, Faculdade de Ciências - Universidade do Porto, Rua do Campo Alegre 687, Porto, PORTUGAL c UI&D Matemática e Aplicações, Universidade de Aveiro, Aveiro, PORTUGAL Abstract The Iteger-valued AutoRegressive (IAR) processes were introduced in the literature by Al-Osh and Alzaid (1987) and McKenzie (1988) for modelling correlated series of counts These processes have been considered as the discrete counter part of AR processes, but their highly nonlinear characteristics lead to some statistically challenging problems, namely in parameter estimation Several estimation procedures have been proposed in the literature, mainly for processes of first order For some of these estimators the asymptotic properties as well as finite sample properties have been obtained and studied This paper considers Yule-Walker parameter estimation for the pth-order integer-valued autoregressive, IAR(p), process In particular, the asymptotic distribution of the Yule-Walker estimator is obtained and it is shown that this estimator is asymptotically normally distributed, unbiased and consistent Key words: IAR Process, Autocovariance distribution, Yule-Walker Estimation, Delta Method 1 Introduction Recently, there has been a growing interest in modelling non-negative integervalued time series and, specially, time series of counts Several models have been proposed and, in particular, the Iteger-valued AutoRegressive, IAR, addresses: ims@feuppt (Isabel Silva), mesilva@fcuppt (M Eduarda Silva) Preprint submitted to Elsevier Science 3rd April 2006

2 model has been the subject of study in several papers The pth-order integervalued autoregressive, IAR(p), process is defined as follows (Latour, 1998) A discrete time non-negative integer-valued stochastic process, {X t }, is said to be an IAR(p) process if it satisfies the following equation X t α 1 X t 1 + α 2 X t α p X t p + e t, (1) where (1) {e t }, designated the innovation process, is a sequence of independent and identically distributed (iid) non-negative integer-valued random variables with Ee t ] µ e, Vare t ] σ 2 e, Ee 3 t] γ e and Ee 4 t] κ e ; (2) the symbol represents the thinning operation (Steutel and Van Harn, 1979; Gauthier and Latour, 1994), defined by α i X t i X t i j1 Y i,j, i 1,,p, where {Y i,j }, designated the counting series, is a set of iid non-negative integer-valued random variables such that EY i,j ] α i, VarY i,j ] σi 2, EYi,j] 3 γ i and EYi,j] 4 κ i All the counting series are assumed independent of {e t }; (3) 0 α i < 1, i 1,,p 1, and 0 < α p < 1 Alternatively, an IAR(p) process can be represented as a p-dimensional IAR(1) process (Franke and Subba Rao, 1995) Accordingly, by using the vector thinning operation, defined as a random vector whose ith component is given by (Franke and Subba Rao, 1995) A X] i p j1 a ij X j, i 1,,p, where X X 1 X p ] T is a random vector, A is a p p matrix with entries a ij satisfying 0 a ij 1, for i,j 1,,p, and the counting series of all a ij X j, i,j 1,,p, are assumed independent, the IAR(p) process defined in (1) can be written as X t A X t 1 + W t, X t HX t, (2a) (2b) where H ], X t X t X t 1 X t p+1 ] T, W t e t 0 0 ] T, for {e t } a sequence of iid random variables, with Ee t ] µ e, Vare t ] σ 2 e, Ee t 3 ] γ e, and Ee 4 t] κ e <, and 2

3 α 1 α 2 α p 1 α p A Furthermore, the model (2a) can be expressed as X t d A k X t k + k 1 j0 Aj W t j, (3) where d stands for equality in distribution Then, equations (2) can be written as X t d j0 Aj W t j, (4a) X t d HX t, (4b) where H ], provided the spectral radius of the matrix A is less than one, that is, ρ(a) < 1 The existence and stationarity condition for the IAR(p) processes is that the roots of z p α 1 z p 1 α p 1 z α p 0 lie inside the unit circle (Du and Li, 1991) or, equivalently, that p j1 α j < 1, (Latour, 1997, 1998) Probabilistic characteristics of the IAR models, in terms of second and third-order moments and cumulants, have been obtained by Silva and Oliveira (2004, 2005) In particular, it is found that the autocovariance function, R( ), satisfies a set of Yule-Walker type difference equations, which can be written in scalar and vectorial form as R(0) V p + p α ] i1 ir(i), T R(k) R p p α α V p 0 0, i1 ir(i k), respectively, where and V p is defined by R(0) R(1) R(p) 1 R(1) R(0) R(p 1) α R p, α 1, R(p) R(p 1) R(0) α p 3

4 V p σ e 2 + µ X p i1 σ i 2, where σ e 2 Vare t ], µ X EX t ] µ e /(1 p i1 α i ), and σ i 2 is the variance of the counting series, Y j,i (j 1,,X t i ; i 1,,p), of the ith thinning operation, α i X t i Several estimation methods for the IAR(p) model have been proposed in the literature both in the time domain, namely, Yule-Walker and Conditional Least Squares (Du and Li, 1991; Latour, 1998), Generalized Method of Moments (Brännäs, 1994, 1995), Conditional Maximum Likelihood (Franke and Seligmann, 1993; Franke and Subba Rao, 1995) and in the frequency domain, the Whittle Criterion (Silva and Oliveira, 2004, 2005) Methods based on higher-order statistics (moments and cumulants) have also been considered (I Silva, 2005) The small sample properties of these estimators have been studied empirically by Monte Carlo methods (I Silva, 2005) but little is known about the asymptotic properties It must be said that these depend on higher order moments and cumulants which are very difficult to obtain Here the asymptotic distribution of the sample autocovariance for IAR(p) processes is obtained and it is shown that the Yule-Walker estimator is asymptotically normally distributed, unbiased and consistent This result generalizes the work of Park and Oh (1997), who derived the asymptotic distribution of the Yule-Walker estimator for an alternative parametrization of the Poisson IAR(1) process with binomial thinning operation In Section 21 the asymptotic properties of the sample mean and sample autocovariance matrix are obtained and in Section 22 expressions for the asymptotic covariance matrix of the Yule-Walker estimator are provided 2 Yule-Walker Estimation of the IAR(p) model Given a realization {X 1,,X } from an IAR(p) process, the Yule-Walker estimator ˆα of α is obtained by solving the following system of linear equations, ˆR(0) ˆR(1) ˆR(2) ˆR(p 1) ˆα 1 ˆR(1) ˆR(1) ˆR(0) ˆR(1) ˆR(p 2) ˆR(2) ˆR p 1 ˆα ˆr p ˆR(2) ˆR(1) ˆR(0) ˆR(p 3) ˆR(3), ˆR(p 1) ˆR(p 2) ˆR(p 3) ˆR(0) ˆα p ˆR(p) with ˆR(k) 1 k t1 (X t X)(X t+k X),k Z, and X 1 t1 X t 4

5 The estimators for µ e and σ 2 e are, respectively, ˆµ e X ( 1 p ˆα ) i1 i, ˆσ 2 e ˆV p X p i1 ˆσ2 i, where ˆV p ˆR(0) p ˆα ˆR(i) i1 i and ˆσ i 2 is an estimator of the variance of the counting series for the i-th thinning operation, α i X t i,i 1,,p The estimation of ˆσ i 2 depends on the distribution of the counting series For instance, in the case of the binomial thinning operation (when the counting series are Bernoulli distributed), ˆσ i 2 ˆα i (1 ˆα i ), for i 1,,p ow, in order to obtain the asymptotic distribution of the Yule-Walker estimator, the asymptotic properties of the sample covariance matrix are needed These properties are obtained in 21 and the asymptotic distribution of the Yule-Walker estimator follows in Asymptotic distribution of the autocovariance function In the following theorem the asymptotic multivariate normality of the sample mean, X, and of the sample autocovariance function, ˆR(k), is established The details of the proof are omitted since it follows closely Brockwell and Davis (1991, Chap 7) by considering an auxiliary function R (k) 1 t1 (X t µ X )(X t+k µ X ), k 0, obtaining its asymptotic properties and proving that ˆR( ) has the same asymptotic properties as the auxiliary function, R ( ) Theorem 1 If {X t } is an IAR(p) process with representation (4), where j0 A j <, and if R( ) is the autocovariance function of {X t }, then for any non-negative integer h, X ˆR(0) ˆR(h) µ X R(0) is A, 1 V R R(h) where V R V 11] 1 1 V 12 ] 1 (h+1) V 12 ] T 1 (h+1) V 22] (h+1) (h+1) 5

6 is an (h + 2) (h + 2) matrix for any non-negative integer h, such that V 11 lim Var(X) R(h), (5a) h V 12 ] k+1 lim Cov(X,R (k)), k 0, 1,,h, C h X(h,k + h), (5b) V 22 ] k+1,j+1 lim Cov(R (k),r (j)), k,j 0, 1,,h, R(h)R(h + j k) + R(h + j)r(h k) h + C Y (h,k,j + h), (5c) where C X (, ) is the third-order cumulant of X t and C Y (,, ) is the fourthorder cumulant of the process Y t X t µ X (I Silva, 2005) PROOF ote that for 0 k h, (R (k) ˆR(k)) 1 t k+1 (X t µ X )(X t+k µ X ) + 1 ( k µ X X k t1 t µ X X t1 t+k ( k)µ 2 X + X k t1 X t + X k t1 X t+k ( k)x 2 1 t k+1 (X t µ X )(X t+k µ X ) ) + ( 1 k (X µ X ) X t1 t + 1 ( ) ) 1 k (X + µ X ) k X t1 t+k The first term, 1 t k+1 (X t µ X )(X t+k µ X ), is o p (1), since 1 ] E (X t k+1 t µ X )(X t+k µ X ) The last term is also o p (1), since (X µx ) is O p (1), 1 kr(0) 0 as, because (X µ X ) d Y, where Y (0,V 11 ), and d stands for convergence in distribution ote that by the weak law of large numbers (Brockwell and Davis, 1991, p 208), ( 1 k X t1 t + 1 ( k X t1 t+k 1 k ) ) P (X + µ X ) 0 6

7 This remark leads to the following result (R (k) ˆR(k)) is o p (1), as Then the proposition follows from the fact that X µ X R (0) R(0) is A, 1 V R R (h) R(h) and Proposition 633 of Brockwell and Davis (1991, p 205) ote that this result is in agreement with the asymptotic distribution obtained by Park and Oh (1997) for the case of Poisson IAR(1) models with binomial thinning operation 22 Asymptotic distribution of the Yule-Walker estimator In this section, the asymptotic properties of the sample covariance matrix is used to obtain the asymptotic distribution of the Yule-Walker estimator Hence, consider the p p Toeplitz sample autocovariance matrix, ˆR p 1, and the vector of sample autocovariance function with p elements, ˆr p Let ˆV Rr vec( ˆR p 1 ) (6) ˆr p be a vector with p(p + 1) elements, where the vec operator is the columnstacking operator of a matrix (eudecker, 1969), that is, vec(a) A 1 A 2 is a pq 1 vector where A i is the i-th column of the p q matrix A A q ote that for p 1, ˆV Rr ] T ˆR(0) ˆR(1) 7

8 and for p > 1, the i-th element of ˆV Rr can be written as ˆR( (i 1) mod p (i 1)/p] ), if i p ˆV 2, Rr (i) ˆR( i mod p ), if i > p 2, where a] represent the integer part of a R Then, using the previous results it is found that ˆV Rr vec( ˆR p 1 ) is A vec(r p 1), 1 ˆr p r p Σ Rr, where Σ Rr is the p(p + 1) p(p + 1) covariance matrix of ˆV Rr, defined as in (5c), for i,j 1,,p(p + 1) ow, let Σ Rr (i,j) Cov( ˆV Rr (i), ˆV Rr (j)), (7) ] ((Rp 1 D T R(1)I p R(p)I 1 p ) T R ) ] 1 p 1 I p 2 0 p 2 p ] p p 2 R p 1, (8) where I n is the n n identity matrix, 0 n m the n m matrix of zeros and the Kronecker product (Graham, 1981) defined as follows Let A and B be p q and m n matrices, respectively Then A B is a pm qn matrix given by a 11 B a 12 B a 1q B a A B 21 B a 22 B a 2q B a p1 B a p2 B a pq B The asymptotic distribution of the Yule-Walker estimator of an IAR(p) process is given in the following theorem Theorem 2 Let {X t } be an IAR(p) process, satisfying (1), and ˆα the Yule- Walker estimator of α Then 1/2 (ˆα α) is A(0 p,d T Σ Rr D), where 0 n is a vector of n zeros, Σ Rr is given in (7) and D T is defined by (8) 8

9 PROOF The Delta method (van der Vaart, 1998) is used to demonstrate the result Thus, let ϕ be the function from D ϕ R p(p+1) into R p defined by ϕ(x) (unvec p p+1 (X)C 1 ) 1 (unvec p p+1 (X)C 2 ), where X is a vector with p(p + 1) elements, C 1 I p 0 T p, C 2 0 p 1 are a (p + 1) p matrix and a vector with p + 1 elements, respectively, 0 p is a vector of p zeros and unvec n m is the inverse of the vec operator, that is, unvec p q (X), for a vector X of pq 1 elements is defined as a p q matrix, such that unvec p q (vec(a)) A (Swami and Giannakis, 1994) ote that for ˆV Rr defined in (6), ϕ( ˆV Rr ) ˆR 1 p 1ˆr p ˆα Then, by the Delta method, ˆα is A ( α, 1 ) DT Σ Rr D, where ( ) ( ϕ ϕ D X X ˆV Rr ) T X ˆV Rr is the p(p + 1) p derivative matrix of the function ϕ, defined as follows Let f : R n R m be a vector valued function with vector variable The n m matrix derivative of f is defined as (Rao and Rao, 1998, p 225) ( ) T f f X f 1 f X 1 1 X n f m X 1 f m X n in order to meet the needs of the Jacobian matrix ote that ϕ(x) can be written as the product of two matrices, ϕ(x) M 1 1 (X)M 2 (X) Then, by using the product rule (Rao and Rao, 1998, p T, 9

10 234), it is found that ( ϕ ) T ) T ( 1 (M1 M 2 ) ( (M T 2 I p ) vec(m 1 1 ) + (I 1 M 1 1 ) vec(m ) T 2) ow, using the matrix derivative of an inverse matrix, vec(m 1 1 ) ( (M 1 1 ) T M 1 1 ) vec(m 1 ), and then, replacing M 1 by unvec p p+1 (X)C 1, and using the properties of the vec and unvec n m operators (eudecker, 1969) and the derivative matrix rule of f : R k R such that f AX, where A is a n k constant matrix (Rao and Rao, 1998, p 233), it is found that vec(m 1 1 ) ( ((unvec p p+1 (X)C 1 ) 1 ) T (unvec p p+1 (X)C 1 ) 1) vec(unvec p p+1 (X)C 1 ) ( ((unvec p p+1 (X)C 1 ) 1 ) T (unvec p p+1 (X)C 1 ) 1) In a similar manner Then, (C 1 T I p )X ( ((unvec p p+1 (X)C 1 ) 1 ) T (unvec p p+1 (X)C 1 ) 1) (C 1 T I p ) vec(m 2 ) vec(unvec p p+1(x)c 2 ) (C 2 T I p )X (C T 2 I p ) ( ϕ Therefore, ) T ( ((unvec p p+1 (X)C 2 ) T I p ) ( ((unvecp p+1 (X)C 1 ) 1 ) T (unvec p p+1 (X)C 1 ) 1) (C 1 T I p ) + (I 1 (unvec p p+1 (X)C 1 ) 1 )(C 2 T I p )) T D T (r p T I p )((R p 1 1 ) T R p 1 1 )(C 1 T I p ) + (I 1 R p 1 1 )(C 2 T I p ) 10

11 ] ( ) ] R(1)I p R(p)I p (R 1 p 1 ) T 1 R p 1 I p 2 0 p 2 p ] p p 2 R p 1, and the theorem follows For the particular case of IAR(1) processes with Poisson, P(λ), ] innovations ] and binomial thinning operation, it is found that R 0 R(0), Var( ˆR(0)) Cov( ˆR(0), ˆR(1)) Σ Rr Cov( ˆR(0), ˆR(1)) Var( ˆR(1)) λ(2λ(1+α 2 )+(1 α)(1+α) 2 ) (1 α) 2 (1 α 2 ) 2λα(1+2λ α 2 ) (1 α) 2 (1 α 2 ) 2λα(1+2λ α 2 ) (1 α) 2 (1 α 2 ) λ(λ(1+4 α 2 α 4 )+α(1 α)(1+α) 2 ) (1 α) 2 (1 α 2 ) λ (1 α) and D T ] ( R(1) R(1) R(0) 2 1 R(0) ] 1/R(0) ] α(1 α) λ ]) 1/R(0) 1 α λ ] ] ] /R(0) Hence, D T Σ Rr D (1 α 2 α(1 α)2 ) + Therefore, Theorem 2 states that λ ( ) 1/2 (ˆα α) is A 0, (1 α 2 α(1 α)2 ) +, λ which is in agreement with the asymptotic distribution obtained by Park and Oh (1997) for the α parameter References Al-Osh, MA and Alzaid, AA (1987) First-order integer-valued autoregressive (IAR(1)) process Journal of Time Series Analysis, Vol 8, pp Brännäs, K (1994) Estimation and testing in integer-valued AR(1) models Umeå Economic Studies, 335, Umeå University Brännäs, K (1995) Explanatory variables in the AR(1) count data model Umeå Economic Studies, 381, Umeå University Brockwell, PJ and Davis, RA (1991) Time Series: Theory and Methods, 2nd edition, Springer-Verlag, ew York 11

12 Du, Jin-Guan and Li, Yuan (1991) The integer-valued autoregressive (IAR(p)) model Journal of Time Series Analysis, Vol 12, pp Franke, J and Seligmann, T (1993) Conditional maximum likelihood estimates for IAR(1) processes and their application to modelling epileptic seizure counts In Developments in Time Series Analysis (ed T Subba Rao) Chapmann and Hall, London pp Franke, J and Subba Rao, T (1995) Multivariate first order integer valued autoregressions Technical Report, Math Dep, UMIST Gauthier, G and Latour, A (1994) Convergence forte des estimateurs des paramètrés d un processus GEAR(p) Annales des Sciences Mathématiques du Québec, Vol 18, pp Graham, A (1981) Kronecker products and matrix calculus: with applications Ellis Horwood, Chichester Latour, A (1997) The Multivariate GIAR(p) process Advances in Applied Probability, Vol 29, pp Latour, A (1998) Existence and stochastic structure of a non-negative integer-valued autoregressive process Journal of Time Series Analysis, Vol 19, pp McKenzie, E (1988) Some ARMA models for dependent sequences of Poisson counts Advances in Applied Probability, Vol 20, pp eudecker, H (1969) Some theorems on matrix differentiation with special reference to Kronecker matrix product Journal of the American Statistical Association, Vol 64, pp Park, Y and Oh, CW (1997) Some asymptotic properties in IAR(1) processes with Poisson marginals Statistical Papers, Vol 38, pp Rao, CR and Rao, MB (1998) Matrix algebra and its applications to statistics and econometrics World Scientific Singapore Silva, I (2005) Contributions to the analysis of discrete-valued time series PhD Thesis Universidade do Porto, Portugal Silva, ME and Oliveira, VL (2004) Difference equations for the higher order moments and cumulants of the IAR(1) model Journal of the Time Series Analysis, Vol 25, pp Silva, ME and Oliveira, VL (2005) Difference equations for the higher order moments and cumulants of the IAR(p) model Journal of the Time Series Analysis, Vol 26, pp Steutel, F W and Van Harn, K (1979) Discrete analogues of selfdecomposability and stability The Annals of Probability, Vol 7, pp Swami, A and Giannakis, G (1994) Multichannel ARMA processes IEEE Transactions on Signal Processing, Vol 42, pp van der Vaart, AW (1998) Asymptotic Statistics Cambridge Series in Statistical and Probabilistic Mathematics Cambridge University Press, Cambridge 12