Summary of the Ph.D. Thesis

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1 Summary of the Ph.D. Thesis Katalin Varga Institute of Mathematics and Informatics Debrecen University Pf. 2, H 4 Debrecen, Hungary Introduction In the dissertation stable and nearly unstable multidimensional AR models are studied. Discrete and continuous time process are treated. We study the asymptotic behaviour of the parameter estimators. We give a multidimensional generalization of the famous result of Arató, Kolmogorov, Sinay on the exact distribution of the period of the complex AR model. These processes are well studied in the literature and have been applied as a model for describing random phenomena, such as for instance the behaviour of stoc maret prices. 2 Parameter Estimation with Exact Distribution for Multidimensional Ornstein-Uhlenbec Processes Consider the complex-valued stationary autoregressive process ξ(t) = ξ (t) + iξ 2 (t), t, given by the stochastic differential equation (SDE) dξ(t) = γξ(t)dt + dw(t), where w(t) = w (t) + iw 2 (t), t is a standard complex Wiener process ( ie. w (t) and w 2 (t) are independent standard real valued Wiener process) and γ = λ iω with λ >, ω R. Consider the statistics s 2 ξ(t) = t ξ(u) 2 du, r ξ (t) = t ξ(u) 2 dθ(u), where Θ(t), t is defined by ξ(t) = ξ(t) e iθ(t).

2 It is nown that the maximum lielihood estimate (MLE) of the period ω is ˆω ξ (t) = r ξ(t) s 2 ξ (t), and s ξ (t) (ˆω ξ (t) ω) D = N (, ) for all t, where D = denotes equality in distribution. Surprisingly, we have an exact distribution, not only an asymptotic properly. This result was first formulated and applied in astronomy in Arató, Kolmogorov, Sinay []. Complicated proofs can be found in Noviov [3], Lipster and Shiryayev [4]. Recently Arató [2] gave an elegant new proof using Noviov s method. The following natural question can be formulated. Consider the q-dimensional process X(t), t, given by the SDE dx(t) = AX(t)dt + dw (t), t, where W (t), t is a standard q-dimensional Wiener process and A is a q q matrix. Which conditions should be imposed on the matrix A and on the distribution of the initial value X() in order that the suitably normalised MLE of some of its entries will have exactly a normal distribution? This process is the so-called Ornstein-Uhlenbec process and considered as a generalization of Wiener processes. Pap [6] and Fazeas [5] found some examples for stationary multidimensional Ornstein- Uhlenbec processes which have the above property. Pap and van Zuijlen gave a multidimensional generalization. We could weaen the condition of Pap and van Zuijlen by showing that a part of the conditions is superfluous in [7]. Consider the following multidimensional process: m dx(t) = ( λi d + ω i C i )X(t)dt + dw (t), X() = () i= where I d is the d d unit matrix, λ, ω,..., ω m R are unown parameters and C,..., C m are fixed sew-symmetric matrices. The MLE of ω = (ω,..., ω m ) T is given by ˆω X (t) = σ X (t)r X(t). where σ x (t) is the m m matrix ( t ) σ x (t) = C i X(s), C j X(s) ds, ( i, j m) and r X (t) is the m-dimensional column vector ( t T r X (t) = C i X(s), dx(s) ), ( i, m). In [7] it is proved that σ /2 X (t)(ˆω X(t) ω) D = N (, I m ) for all t >, (2) 2

3 if conditions (C) (C3) are satisfied, where (C) Ci T = C i, i =,..., m (C2) (C i C j + C j C i )C = C (C i C j + C j C i ), i, j, =,..., m (C3) (C i C j + C j C i )(C C l + C l C ) L(C u C v, u, v m), i, j,, l =,..., m where L(C u C v, u, v m) denotes the linear hull of the matrixes C u C v, u, v m. We showed that condition (C3) is superflous. Theorem Let X(t), t, be the process given by (). Let us suppose that the conditions (C) and (C2) satisfiesd. Then (2) holds. 3 Nearly Unstable AR Models with Coefficient Matrices in Jordan Normal Form Consider the d-dimensional autoregressive model { X = QX + ε, =, 2,..., X =, where the d dimensional random column vector ε contains the unobservable random innovation at time, and the d d matrix Q is the unnown parameter of the model. The least squares estimator (LSE) of Q based on the observations X,..., X n is given by ( n ) ( n Q n = X X X X ). = Let ϱ(q) denote the spectral radius of the matrix Q, i.e., the maximum of the absolute values of the eigenvalues of the matrix Q. Asymptotically stationary models, when ϱ(q) <, were studied by Mann and Wald [8] and Anderson [9]. Under the assumption that the ε s are i.i.d. with Eε =, Eε ε = Σ, the LSE of Q is assymptotically normal: where ( n ) /2 ( Q n Q) X X D N d d (, I), as n, = D denotes convergence in distribution and I is the unit matrix. When ϱ(q) =, the model is said to be unstable. The one-dimensional unstable AR() model X = QX + ε,, with Q = was studied by White [] and it was shown that the variables n( Q Q) converge in law to a random variable: n( Q Q) D = W (t) dw (t) W, 2 (t) dt where {W (t), t } is a standard Wiener process. Multidimensional unstable models are studied in Sims, Stoc and Watson [], Tsay and Tiao [2], and in Arató [3]. 3

4 These results led to the study of the following so called nearly unstable models. Nearly unstable or nearly nonstationary multidimensional AR processes are generated according to the scheme { X (n) = Q n X (n) X (n) =, + ε(n), =, 2,..., n, where {ε (n) } is an array of d dimensional random vectors and Q n, n, is a sequence of d d matrices such that Q n Q, where Q is a matrix with ϱ(q) =. The case when Q n = e A/n, n, where A is a fixed d d matrix was studied by Phillips [7]. Kormos and Pap [8] treated the case when Q n = e (γi+a)/n, n, where γ R and A is a sew symmetric matrix under the assumption that ε s are i.i.d. variables. Stocmarr and Jacobsen [4] investigated the case when Q n = I + A/n. Pap and Zuijlen [5] studied the case when Q n = e An/n e B, n, where A n A, B is a nown sew symmetric d d matrix, and A n B = BA n, n. Pap and van Zuijlen [] also studied the case when the model is complex valued and the coefficient matrices are in Jordan normal form. Our aim was to investigate nearly unstable complex valued models, where the coefficient matrices are in Jordan normal form and to study the limit behaviour of the suitable normalised LSE of the eigenvalue. It will turn out that the limit distribution depends only on the last, d th component of the process. We also compared it with the maximum lielihood estimator (MLE) of the eigenvalue of the coefficient matrix of the related continous time model. It is interesting to note that the MLE of the eigenvalue in the related continous time model depends only on the first coordinate of the process. Matrices consisting of two or more Jordan blocs were also studied. (3) 3. Least Squares Estimator of the Eigenvalue Let C d be the space of the d dimensional complex column vectors. Let us introduce the widely used notation x, y := x y for x, y C d for the scalar product in C d, where y denotes the complex conjugate of y. We introduce the norm of x C d by x := x, x. For λ C and d N we introduce the notation λ λ J(λ, d) := λ λ λ for a d d matrix in Jordan normal form with eigenvalue λ. We shall use the short notation J(λ) it it does not cause misunderstanding. 4

5 Consider the d dimensional complex valued autoregressive model { X = J(λ)X + ε, =,2,..., X =, where λ C is the unnown parameter of the model. If we tae into consideration the special form of the coefficient matrix then we can calculate the LSE of λ as follows. Lemma. The LSE of λ, based on the observations X,..., X n is given by λ := n = d j= X,j X,j, X,j n = d j= X,j 2, where X = (X,,..., X,d ) and X, :=. 3.2 Convergence of the LSE For n =, 2,... consider the d dimensional complex valued AR model: { X (n) = J(λ n )X (n) + ε(n), =, 2,..., n, X (n) =, (4) where {ε (n) } is an array of random vectors in Cd and λ n = e hn/n+iθ with h n C, h n h C and θ ( π, π]. Clearly J(λ n ) J(e iθ ) since λ n e iθ. It is easy to see that the model is nearly unstable, since ϱ(j(e iθ )) =. Consider the d dimensional complex valued nearly unstable AR process generated according to the scheme (4). We are interested in the limit behaviour of the LSE of λ n which has the following form according to Lemma : λ n = n = d j= X(n),j X(n),j, X(n),j n = d j= X(n),j 2, where X (n), :=. Theorem Suppose that array ε (n) =,..., n, n statisfies the condition (C). Then n d D ( λ n λ n ) e (d )θi Y d(t) dw d (t) Y d(t) 2 dt, (5) where the process Y (t), t [, ] is given by dy (t) = AY (t)dt + ΣdW (t) Y () =, and W (t), t [, ] is a standard complex d dimensional Wiener process. 5

6 Condition C ε (n), =,..., n, n, is a triangular array of square integrable martingale differences in C d with respect to the filtrations (F (n) ) =,,...,n;n such that for all t [, ] where η (n) nt [nt] = ( ( E η (n) η (n) ) F (n) := (Re (ζ (n) ), Im (ζ(n) )) and ) P 2 I 2d as n, α > n [nt] = E( ε (n) 2 χ (n) { ε (n) F >α n} ) P as n. 4 Estimation of the Mean of Multivariate AR Processes Consider a stationary AR() process { X(t) : t R} which is the wealy stationary solution of d X(t) = α X(t) dt + dw (t), t R, (6) where {W (t) : t R} is a standard Wiener process (hence EW (t) =, EW (t) 2 = t ), and α > is the damping parameter. Let Z(t) := X(t) + m, t R, where m R is an unnown parameter. Then, the maximum lielihood estimator (MLE) of m based on the observation of { Z(t) : t [T, T 2 ]} is given by m = Z(T ) + Z(T 2 ) + α T 2 T 2 + (T 2 T )α Z(t) dt N ( ) m,, (7) 2α + (T 2 T )α 2 see, e.g., Arató [2]. We have the following asymptotic behaviour of the variance of m lim (T 2 T )Var m = T 2 T α, 2 lim α αvar m = 2. Especially, lim Var m =, lim T 2 T Var m =, α hence m is asymptotically consistent as T 2 T, although not uniformly in α. Furthermore, Var m is unbounded as α. The number of parameters can be reduced in the following way. We may suppose T = and T 2 = T because of the stationarity. Moreover, let us consider the process { Z(t) := Z(T t)/ T : t [, ]}. Then Z(t) = X(t) + m/ T, where X(t) := X(T t)/ T, hence d X(t) = κ X(t) dt + dw (t), t R, 6

7 where κ := αt is a new parameter. Considering a := m/ T as a new parameter, we obtain that the MLE of a based on the observation of { Z(t) : t [, ]} is given by ã = Z() + Z() + κ 2 + κ see Arató [2]. For the variance of ã we have Especially, lim κ κ2 Var ã =, lim Var ã =, lim κ Z(t) dt N ( ) a,, (8) 2κ + κ 2 lim κvar ã = κ 2. Var ã =, κ hence ã is asymptotically consistent as κ, but Var ã is unbounded as κ. Our aim was to investigate the problem of estimation of the mean for stationary and zero start multidimensional autoregressive processes. 4. Case of Stationary AR Process Consider a d dimensional stationary AR process {Ỹ (t) : t R}, which is the wealy stationary solution of dỹ (t) = AỸ (t) dt + Σ dw (t), t R, (9) where A and Σ are d d matrices such that all eigenvalues of A have necessarily positive real part, and {W (t) : t R} is a d dimensional standard Wiener process, i.e., {W (t) : t } and {W ( t) : t } are independent d dimensional standard Wiener processes. In fact, {W (t) : t R} is a continuous zero mean Gaussian process with { Cov (W (s), W (t)) = EW (s)w (t) ( s t ) I, if st, =, otherwise, or, equivalently, {W (t) : t R} is a continuous process with stationary independent increments such that for the increments we have W (t) W (s) N d (, t s I). The process {Ỹ (t) : t R} is a zero mean Gaussian process with Cov (Ỹ (t), Ỹ (s)) = EỸ (t)ỹ (s) = e (t s)a B(), for s t, where the d d matrix B() is the only solution of the matrix equation AB() + B()A = ΣΣ. We remar that B() is a symmetric, positive definite matrix and B() = e ua ΣΣ e ua du. 7

8 Moreover, Ỹ (t) = e ta (Ỹ () + t ) t e ua Σ dw (u) = e (t u)a Σ dw (u). () Now, consider a shifted stationary AR process { Z(t) : t R}, given by Z(t) = Ỹ (t) + Hm, t R, where H is a nown d d constraint matrix and m R d is an unnown parameter vector. Definition For a d d matrix A, we denote by A the pseudo inverse of A which is the unique d d matrix satisfying the following properties.. AA A = A, 2. There exist matrices U and V such that A = UA = A V. Theorem Let [T, T 2 ] R. If H exists then the maximum lielihood estimator of m based on the observation of { Z(t) : t [T, T 2 ]} is given by where m = U V ez, () U = H B() H + (T 2 T )H A ( ΣΣ ) AH, V ez = H B() Z(T ) + H A ( ΣΣ ) ( Z(T 2 ) Z(T 2 ) + A T2 T ) Z(t) dt. Moreover, m N d (m, U ). (2) 5 Case of Zero Start AR Process Consider a d dimensional zero start AR process { dy (t) = AY (t) dt + Σ dw (t), t, Y () =, (3) where A and Σ are arbitrary d d matrices. The process {Y (t) : t } is a zero mean Gaussian process with Cov (Y (t), Y (s)) = EY (t)y (s) = e (t s)a B(, s), for s t, 8

9 where the d d matrix B(, s) is the only solution of the matrix equation AB(, s) + B(, s)a = ΣΣ e sa ΣΣ e sa. We remar that B(, s) is a symmetric, positive definite matrix and Moreover, B(, s) = Y (t) = s t e ua ΣΣ e ua du. e (t u)a Σ dw (u). Now, consider a shifted stationary AR process {Z(t) : t }, given by Z(t) = Y (t) + m, t, where m R d is an unnown parameter vector. The following theorem can be proved as in the stationary case. Theorem Let < T < T 2. The maximum lielihood estimator of m based on the observation of {Z(t) : t [T, T 2 ]} is given by where m = U V Z, (4) U = B(, T ) + (T 2 T )A ( ΣΣ ) A, V Z = B(, T ) Z(T ) + A ( ΣΣ ) ( Z(T 2 ) Z(T )) + A T2 T ) Z(t) dt. Moreover, m N d (m, U ). (5) We mention if T = then Z() = Y () + m = m gives the precise value of the parameter m. References [] Arató, M., Kolmogorov, A.N. and Sinay, Ya. G. (962). Estimation of the parameters of a complex stationary Gaussian Marov Process, Dol. Aad. Nau SSSR 46, [2] Arató, M. (982). Linear stochastic systems with constant coefficients. A statictical approach. (Lecture Notes in Control and Inf., vol. 45, 39 pp.) Springer-Verlag, Berlin (in Russian, Naua, Moscow, 989). 9

10 [3] Noviov, A. (972) On the estimation of parameters of diffusion processes. Studia Sci. Math. Hungarica 7, [4] Liptser, R.S. and Shiryayev, A.N. (977). Statistics of Random Processes I, II, Springer-Verlag, New Yor. [5] Fazeas, I. (993). On maximum lielihood estimation of parameters of multidimensional stationary AR processes. Technical Report, Dept. of Math., Lajos Kossuth Univ., Debrecen, 84/993. [6] Pap, G. (994). On the distribution of estimates of parameters of multidimensional stationary AR processes, Computers Math. Appl. 27, 8. [7] Pap, G. and van Zuijlen, M.C.A. (993). Parameter estimation with exact distribution for multidimensional Ornstein-Uhlenbec process, Report 934, Dept. of Math. Catholic Univ. Nijmegen. [8] H.B. Mann and A. Wald, On the statistical treatment of linear stochastic difference equations, Econometrica, (943). [9] T.W. Anderson, The statistical analysis of time series, John Wiley & Sons, New Yor, London, Sydney, Toronto, (97). [] J.S. White, The limiting distribution of the serial correlation coefficient in the explosive case, Ann. Math. Statist. 29, (958). [] C.A. Sims, J.H. Stoc and M.W. Watson, Inference in linear time series models with some unit roots, Econometrica 58, 3 44 (99). [2] R.S. Tsay and G.C. Tiao, Asymptotic properties of multivariate nonstationary processes with applications to autoregressions, Ann. Statist. 8, (99). [3] M. Arató, Asymptotic inference for discrete vector AR processes, Publ. Math. 36, 9 3 (989). [4] A. Stocmarr and M. Jacobsen, Gaussian diffusions and autoregressive processes: wea convergence and statistical inference, Scan. J. Statist. 2, (994). [5] Pap, G. and van Zuijlen, M.C.A. Asymptotic properties of nearly unstable multivariate AR processes, to appear in Computers Math. Appl. (998).