On Statistical Problems of Discrete and Continuous Time Autoregressive Processes

Átírás

1

2 2

3 On Statistical Problems of Discrete and Continuous Time Autoregressive Processes Katalin Varga Institute of Mathematics and Informatics University of Debrecen, Hungary 23 Tutor: Gyula Pap

4 2 Ezen érteezést a Debreceni Egyetem Matematia dotori program Valószín ségelmélet és matematiai statisztia alprogramja eretében észítettem özött és ezúton benyújtom a Debreceni Egyetem dotori PhD foozatána elnyerése céljából Debrecen, 23 július 5 Varga Katalin jelölt Tanúsítom, hogy Varga Katalin dotorjelölt özött a fent megnevezett dotori alprogram eretében irányításommal végezte munáját Az érteezésben foglalta a jelölt önálló munáján alapulna, az eredményehez önálló alotó tevéenységével meghatározóan hozzájárult Az érteezés elfogadását javaslom Debrecen, 23 július 5 Dr Pap Gyula témavezet

5 3 Acnowledgement Here I would lie to than all the people who have contributed to my dissertation To my supervisor, Dr Gyula Pap from whom I have learned and still learn a lot, and who has helped me with his instructions and advice in writing the dissertation To my professors, Dr Mátyás Arató and Dr Gábor Tusnády for the valuable discussions, which widened my sphere of vision

6 4 Köszönetnyilvánítás Itt szeretné öszönetet mondani azona, ai hozzájárulta disszertációm elészítéséhez Témavezet mne, Dr Pap Gyulána, ait l soat tanultam és tanulo ma is, ai tanácsaival és útmutatásával segített abban, hogy ez a dolgozat létrejöhessen Tanáraimna, Dr Arató Mátyásna és Dr Tusnády Gáborna a gondolatébreszt beszélgetéseért, amelye szélesítetté látóörömet

7 Contents Preface 9 2 Preliminaries 5 2 Processes in Discrete Time 5 2 Estimators for Dierent Models 6 22 Processes in Continuous Time 2 22 Radon-Niodym Derivatives and MLE's 2 3 Exact Distribution of the MLE 25 3 MLE of the Multidimensional Case Proof of the Theorem Special Cases 3 4 Nearly Unstable AR Models 33 4 The Shifted AR Process 33 4 Case of Zero Start AR Model Case of Stationary Nearly Unstable AR Model Nearly Unstable Multidimensional AR Processes Nearly Unstable AR Models with Coecient Matrices in Jordan Normal Form Convergence of a Related Step Process 48 5

8 6 CONTENTS 423 Convergence of the LSE MLE of the Coecient of the Related Continuous Time Model The Case of Several Jordan Blocs 55 5 Estimation of the Mean 57 5 Case of Stationary AR Process 57 5 Examples of Stationary Processes Case of Zero Start AR Process 7 52 Examples of Zero Start Processes 7 Összefoglaló Hungarian Summary 75 A List of Publications 83 B Conference Tals 85

9 List of Notations N set of natural numbers R set of real numbers R d d-dimensional Euclidean space C set of complex numbers x the absolute value of x x the norm of x in a linear space x complex conjugate of x, x C d Re x the real part of the complex number x Im x the imaginary part of the complex number x <, > scalar product E expectation EX Y conditional expectation of X given Y A transpose of matrix A TrA trace of matrix A DetA determinant of A ϱa spectral radius of matrix A A pseudo inverse of matrix A I d d d identity matrix Ψ T α Laplace transform ˆQ estimator for Q D MLE LSE P X wea convergence maximum lielihood estimate least squares estimate the measure generated by process X normal distribution with mean vector m and covariancia matrix D Radon-Niodym derivative N m, D dp X dp W x D = equality in distribution SDE stochastic dierential equation 7

10 8 CONTENTS

11 Chapter Preface Why linear stochastic models? The idea of using mathematical models to describe the behaviour of physical, economical phenomenon is well established In particular, it is sometimes possible to derive a model based on physical laws, which enables us to calculate the value of some time-dependent quantity nearly exactly any instant of time If such exact calculations were possible the model would be entirely deterministic In many problems we have to consider a time-dependent phenomenon, in which there are many unnown factors and for which it is not possible to write a deterministic model that allows exact calculations of the future behaviour of the phenomenon Nevertheless, it may be possible to derive a model that can be used to calculate the probability of the future values lying between two specied limits Such a model is called stochastic model Among stochastic models the class of linear models is wide enough to describe many important empirical time series and can be handled well mathematically In a linear stochastic model the time series is supposed to be linear aggregation of random shocs For practical representation it is desirable to employ models which use parameters parsimoniously Parsimony can be achieved by representation of the linear process in terms of a small number of autoregressive and moving average terms In these models the current value of the process is expressed as a nite, linear aggregate of previous values of the process and random noise Since 97, when Box and Jenins wored out the theory of ARMA modelling, the importance of these models have been growing continuously An important class of stochastic models for describing time series is the so called stationary models, which assume that the process remains in equilibrium about a constant mean level However, in industry, business and economics there are many time 9

12 CHAPTER PREFACE series, which are often better represented as nonstationary There are an unlimited number of ways in which a process can be nonstationary Nevertheless, many important types of economic and nancial time series frequently exhibit a particular ind of homogeneous nonstationary behaviour, that can be represented by a stochastic model, which is modied form of the autoregressive model The thesis deals with some statistical questions of linear, particularly autoregressive models in discrete and continuous time Stationary and nonstationary time series are discussed as well We study the limit distribution of the estimators of the modelparameters In Chapter 2 there is a summary of the results which we consider as a starting point of the dissertation In Chapter 3, in Chapter 4 and Chapter 5 the new results of the author are presented These chapters are based on one paper of the author and three papers of the author with coauthors In Chapter 3 we treat the generalization of the result of Arató, Kolmogorov, Sinay [6] on the exact distribution of the maximum lielihood estimate MLE of the period of the complex-valued stationary rst order autoregressive process The complex-valued stationary autoregressive process ξt = ξ t + iξ 2 t, t, can be given by the stochastic dierential equation SDE dξt = γξtdt + dwt, where wt = 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 processes and γ = λ iω with λ >, ω R This equation describes the rotation of the instantaneous axis of rotation of the earth with respect to the minor axis of the terrestrial ellipsoid, after the elimination of the one year periodic component, and it is called Chandler wobble Arató in [4] gave a new proof that the suitably normalized MLE of the period ω exactly normally distributed If we rewrite the SDE as a two-dimensional real-valued SDE, the following natural question arises Consider the d-dimensional process Xt, t, given by the SDE dxt = AXtdt + dw t, t, where W t, t is a standard d-dimensional Wiener process and A is a d d matrix Which conditions should be imposed on the matrix A and on the distribution of the initial value X in order to get similar results on the suitably normalized MLE of some entries of the matrix A? Pap and van Zuijlen gave a multidimensional generalization In [] we could weaen the conditions of Pap and van Zuijlen by showing that a part of the conditions is superuous in [27] Consider the following

13 multidimensional process: dxt = λi d + m ω i C i Xtdt + dw t, X = i= where I d is the d d unit matrix, λ, ω,, ω m R are unnown parameters and C,, C m are xed sew-symmetric matrices The MLE of ω = ω,, ω m is given by ˆω X t = σ X tr Xt, where σ X t is the m m matrix t σ X t = C i Xs, C j Xs ds and r X t is the m-dimensional column vector t r X t = C i Xs, dxs i,j m i m In [] we proved that under the weaend conditions, the suitably normalized MLE of the periodic parameter ω is exactly normally distributed In Chapter 4 nearly unstable, nearly stable and explosive AR models are studied A multidimensional AR model can be characterized by the help of its coecient matrix Consider the d-dimensional autoregressive model { X = QX + ε, =, 2,, X =, where the d-dimensional real-valued random vector ε contains the random noise at time, and the d d matrix Q is the unnown parameter of the model Let ϱq denote the spectral radius of the matrix Q, ie, the maximum of the absolute values of the eigenvalues of the matrix Q When ϱq <, the model is said to be stable or asymptotically stationary It is well-now that under the assumption that the ε 's are iid with Eε =, Eε ε = Σ, the least squares estimate LSE of Q is asymptotically normal see [], [22] When ϱq =, the model is said to be unstable or unstationary, when ϱq > the model is explosive The dierent types of models behaves diversely The condition ϱq <, is needed for the process to have a stationary solution In this case the process is damped The explosive model "breas loose" after a short induction period, with the generating noise playing almost no further part The unstable model is also called unit root case and have been the subject of much recent attention in the econometrics literature In part, this is,

14 2 CHAPTER PREFACE because the unit root hypothesis is of considerable interest in applications, not only with data from nancial and commodity marets where it has a long history but also with aggregate time series As we will see later, the estimators of parameters in these models behaves dierently In the rst part of Chapter 4, which is based on [3], we study a sequence of nearly unstable AR models: where { X n = α n X n X n =, + εn, =, 2,, α n = γn n, γn γ R 2 We discuss the questions concerning the LSE of the shift-parameter m n of the model Z n := X n + m n h n, =, 2,, n =, 2,, where m n R is an unnown parameter and h n, =,, n, n =, 2, are nown constraints Our purpose was to investigate the limit behaviour of m n T,T 2 as n and the connection between m n T,T 2 and the MLE m T,T 2 of the shift-parameter of the related continuous time model We discuss these questions in the nearly stable, nearly unstable and nearly explosive cases It turns out that m n T,T 2 is always asymptotically normal, but the speed of convergence is much less in the nearly unstable case than in the unstable case The stationary case is also treated, when m n T,T 2 might not be asymptotically normal if γ n, where γ n is the damping parameter of the model In the second part of Chapter 4, which is based on [2], nearly unstable multidimensional AR processes are treated We discuss models where the coecient matrices are in Jordan normal form Consider the ddimensional complexvalued autoregressive model { X = Jλ, dx + ε, =,2,, X =, where λ C and d N, Jλ, d is a d d matrix in Jordan normal form, with

15 eigenvalue λ: λ λ Jλ, d := λ λ λ and λ is the unnown parameter of the model If the random innovations ε satisfy a usual, rather general condition, then we found that the suitably normalised LSE of the eigenvalue λ n converges wealy to a functional of a special d-dimensional Ornstein-Uhlenbec process: n d λ n λ n where the process Y t, t [, ] is given by D e d θi Y dt dw d t Y dt 2 dt, dy t = AY tdt + ΣdW t Y =, with a certain matrix A, and W t, t [, ] is a standard complex ddimensional Wiener process We also prove that in the one-dimensional model the LSE of the discrete time models converges to the MLE of the coecient of the corresponding continuous time model If d 2 then such relationship does not hold In Chapter 5, which is based on [4], we treat the estimation of the mean of multivariate AR processes In Arató [3] the estimation of the unnown mean of realand complex-valued AR processes is discussed Our aim was to investigate the problem of estimation of the mean for stationary and zero start multidimensional autoregressive processes We show that for autoregressive processes the estimators of the mean are consistent if the component of the process is 'periodical', and it is not the case if the component is a 'damping' one In the one-dimensional AR case, the mean cannot be estimated well In the complex AR, where the process behaves periodically, the mean can be estimated well For an AR2 process, the mean can be estimated well, if the roots of the charasteristic equation are complex It is clear that the present results can be used fruitfully in a statistical context such as in problems of testing hypotheses, estimating parameters, and also in constructing condence regions for the unnown parameters 3

16 4 CHAPTER PREFACE

17 Chapter 2 Preliminaries In this chapter we introduce discrete and continuous time multidimensional AR models and show how one can get estimates of the most important parameters eg the coecient matrix or the shift-parameter 2 Processes in Discrete Time A d-dimensional stochastic vector process X t = X t,, X d t is called elementary Gaussian if it is stationary, Marov and Gaussian In continuous time it is assumed that the process is of diusion type It is supposed that the process Xt is nondegenerate and it is linearly regular or purely non-deterministic Nondegeneracy means that the components of Xt are pointwise linearly independent Xt is called linearly regular if it has no deterministic component In this case the Wold's expansion holds in discrete time, see eg Arató [3], Shiryayev [3] Xt = A εt, = where εt is d-dimensional process, EXt =, Cov εt, εt exists, Cov εt, εt =, Such an εt process will be called a white noise process The connection between stochastic dierence equations and elementary processes can be characterized In the following we denote the time by n in the discrete time case 5

18 6 CHAPTER 2 PRELIMINARIES Let ε n be a d-dimensional Gaussian white noise process with parameters Eε n =, Cov ε n, ε n = B ε, where ran B ε Let Q be a non-singular d d matrix Let us assume that the equation B = QBQ + B ε 2 has a non-singular, symmetric, positive denite solution B The connection is the following, see Arató [3]: Theorem 2 The process X n is a d-dimensional Gaussian process if and only if it is the solution of the following stochastic dierence equation in the following sense: X n = QX n + ε n, n =,, 2,, 22 i Let ε n be a d-dimensional Gaussian white noise independent, identically distributed sequence of random vectors with covariance matrix B ε B ε, Eε n = and let Q be a non-singular d d matrix with eigenvalues in the open unit disc Then equation 22 has a unique regular stationary solution which is a Gaussian Marov process where Cov X n, X n = B is the solution of 2 ii Let X n n =, ±, ±2, be a nondegenerate, linearly regular d-dimensional elementary Gaussian process with EX n =, Cov X n, X n = B, B Then there exists a nonsingular d d matrix Q, with eigenvalues in the open unit disc and a sequence of independent, identically distributed Gaussian vectors ε n such that equation 22 holds Eε n = and B ε is uniquely determined by 2 2 Estimators for Dierent Models Often the practitioner must determine the degree of the autoregressive process as well as estimate the parameters If it is possible to specify a maximum for the degree of the process, a process of that degree can rst be estimated and higher order terms discarded using standard regression statistics Anderson 962 gave a procedure for this decision problem Various other model building methods based on regression theory can be used In the dissertation we don't treat the question of model building We are dealing with estimating the parameters of some specied model The following model is one of the simplest and most heavily used models in time series analysis The vector AR process is given by: X n = QX n + ε n, n =,, 2, 23

19 2 PROCESSES IN DISCRETE TIME 7 The d-dimensional random column vector ε n contains the unobservable random innovation at time n We study the case when 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 ϱq denote the spectral radius of the matrix Q, ie, the maximum of the absolute values of the eigenvalues of the matrix Q Asymptotically stationary models, when ϱq <, were studied by Mann and Wald [22] and Anderson [] Under the assumption that the ε 's are iid with Eε =, Eε ε = Σ, the LSE of Q is assymptotically normal: n /2 Q n Q X X D Nd d, I, as n, where = D denotes convergence in distribution and I is the unit matrix When ϱq =, the model is said to be unstable The one-dimensional unstable AR model X = QX + ε,, with Q = was studied by White [33] 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 [3], Tsay and Tiao [29], and in Arató [4] In the explosive case: { X = αx + ε, =, 2,, X = 2 24 when α >, the sequence α n n is again not asymptotically normal If, for example, ε N,, then α n α n α D Cauchy, α 2 In general, the limit distribution depends on the distribution of ε White [33], Anderson []

20 8 CHAPTER 2 PRELIMINARIES By another normalization, n j= X 2 j /2 α n α which holds in case α > if ε N, N,, if α, D W t dw t /2, if α =, W 2 t dt These results led to the study of the following socalled nearly unstable models Nearly unstable or nearly nonstationary multidimensional AR processes are generated according to the scheme { X n = Q n X n + εn, =, 2,, n, X n 25 =, where {ε n } is an array of ddimensional random vectors and Q n, n, is a sequence of d d matrices such that Q n Q, where Q is a matrix with ϱq = The case when Q n = e A/n, n, where A is a xed 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 sewsymmetric matrix under the assumption that ε s are iid variables Stocmarr and Jacobsen [32] investigated the case when Q n = I + A/n Pap and Zuijlen [26] studied the case when Q n = e An/n e B, n, where A n A, B is a nown sewsymmetric d d matrix, and A n B = BA n, n Pap and van Zuijlen [] also studied the case when the model is complexvalued and the coecient matrices are in Jordan normal form where In the nearly unstable zero start AR model { X n = α n X n + εn, =, 2,, X n =, α n = γn n, γn γ R, 26 the LSE α n n of α n based on the observations {X n : =, 2,, n} has the asymptotic behaviour n α n n α n D Y t dw t Y, 27 2 t dt

21 2 PROCESSES IN DISCRETE TIME 9 where {Y t : t [, ]} is a continuous time AR process, ie, an Ornstein- Uhlenbec process, dened as the solution of the stochastic dierential equation { dy t = γy t dt + dw t, t, 28 Y =, Bobosi [9], Phillips [28], Chan and Wei [] Arató, Kolmogorov and Sinay [6] and Arató [2], [4] has drawn the attention to the connection between discrete and continuous time models The result 27 can also be formulated as γ n n D γ, where is the LSE of γ n γ n n = n n j= Xn j Xn j n j= X n j X n 2 in the discrete time model 26 and j γ = Y t dy t Y 2 t dt is the LSE of γ based on the observations {Y t : t [, ]} in the continuous time model 28, which is also the maximum lielihood estimator MLE of γ see [3], Meer, Pap, Zuijlen [23] The distribution of γ is tabulated in Arató [3] In the dissertation the nearly unstable shifted AR model is also treated The nearly unstable AR model: { X n = α n X n + εn, =, 2,, X n 29 =, where α n = γn n, γn γ R, Now, consider the shifted model Z n := X n + m n h n, =, 2,, n =, 2,, where m n R is an unnown parameter and h n, =,, n, n =, 2, are nown constraints

22 2 CHAPTER 2 PRELIMINARIES 22 Processes in Continuous Time The real observation are realized by discrete time Nevertheless, the time-continuous model has its advantages In many cases it is much better to wor with the timecontinuous model Some phenomena can be described more adequately in that way, in other cases the result have more simpler form The statitical questions of continuous time AR processes are studied in Arató [3] Let W t, F t be a d-dimensional standard Wiener process with local parameters EW t = the drift is, EW tw t = ti with unit diusion parameter Let us consider the linear stochastic dierential equation with the non-singular d d matrix A and, may be singular, positive semidenite matrix B W or in integral form dxt = AXtdt + B 2 W dw t, 2 t Xt = Xt + A Xsds + B 2 W W t W t, t where Xt is normally distributed and independent of F W [t,t], t t In one dimension we can interpret this SDE as a the dierential equation which is describing the motion of a particle under the eect of random collisions in the presence of friction, which is proportional to the speed of the particle This is the so-called Ornstein- Uhlenbec process The following statement is true Theorem 22 The continuous d-dimensional random process Xt is an elementary ie stationary Marov Gaussian process if and only if it is the solution of stochastic dierential equation 2 in the following sense i If Xt is a continuous, EXt =, elementary Gaussian process then there exists a unique d d matrix A with eigenvalues in the left halfplane and a Wiener process W t, Ft X =, EW tw t = B W t, such that 2 holds and Bt = EXs + tx s = e At B, t, 2 where with the solution AB + BA = B W 22 B = e As B W e A s ds

23 22 PROCESSES IN CONTINUOUS TIME 2 ii Let d d matrix A be a non-singular matrix with eigenvalues in the left halfplane, and B W non-negative denite, then the only stationary regular solution of 2 with continuous Xt, < t <, is an elementary Gaussian process Its covariance matrix function has the form 2 with B satisfying 22 In the case when Xt is dened for t >, X and Ft W are independent, X is normally distributed with parameters, B We can conclude from the above theorem that a Gaussian process is elementary if and only if its covariance matrix has the form { Bt, s = EXtX e At s B s, t s s = B t e A s t, t s 22 Radon-Niodym Derivatives and MLE's In the statistical investigation of elementary Gaussian processes with continuous time parameter, similarly to the statistics of independent observations, the maximumlielihood principle plays an important role For this purpose it is important to determine the Radon-Niodym derivative of the measure generated by the process with respect to some standard measure Elementary Gaussian processes with common diusion matrix generate equivalent measures, and these measures are equivalent to the Wiener-measure with the same local matrix of variance, see in Liptser, Shiryayev [2] Let C d [, T ] be the metric space of d-dimensional vector-valued continuous functions on the interval [, T ] with the uniform metric It will be convenient to assume C d [, T ] as direct product of the space C d z [, T ] of d-dimensional continuous functions x = {xt, t T } with the initial condition x = z and the d-dimensional Euclidean space R d We consider a Gaussian Marov process Xt satisfying the stochastic dierencial equation 2 and having f as initial probability density function Let P X be the probability measure on C d [, T ] generated by the above process Xt and P W be the "conditional" product of the d-dimensional Lebesque-measure and the measure generated by the Wiener process on the right hand side of 2 Denition 222 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 i AA A = A, ii There exist matrices U and V such that A = UA = A V

24 22 CHAPTER 2 PRELIMINARIES On contrary to the inverse of A, the pseudo inverse of A always exists Theorem 223 If X satises the SDE 2 then the measures P X and P W equivalent and their Radon-Niodym derivative has the form are dp X xt = fxe R T Cxt,dxt R T 2 Axt,Cxt dt, 23 dp W where C = B W A, and f is the density function of the initial value X, that is the distribution of X is absolutely continuous The value of stochastic integral T xt, dxt can be determined for almost all realizations xt with respect to P W Formula 23 may also be written in the following form dp X xt = fxe Tr[C R T dp W where TrA means the trace of A xtdx t] 2Tr[A C R T xtx t]dt, In the time-continuous case we use the Radon-Niodym derivative of X in the lielihood equation to determine the maximum lielihood estimate of some parameters and we can get the sucient statistics as well if they exist In the dissertation we are dealing with the estimation of the mean in multivariate continuous time AR processes Arató in [3] studies the estimation of the mean in the one-dimensional real and complex models He also studied the case in which both the damping parameter and the mean are unnown in the one-dimensional model Here we summarize his results as a starting point of our study Consider a stationary AR process { Xt : t R} which is the wealy stationary solution of d Xt = α Xt dt + dw t, t R, 24 where {W t : t R} is a standard Wiener process hence EW t =, EW t 2 = t, and α > is the damping parameter Let Zt := Xt + m, t R, where m R is an unnown parameter Then, the maximum lielihood estimator MLE of m based on the observation of { Zt : t [T, T 2 ]} is given by m = see, eg, Arató [3] ZT + ZT 2 + α T 2 T 2 + T 2 T α Zt dt N m, 2α + T 2 T α 2, 25

25 22 PROCESSES IN CONTINUOUS TIME 23 Since the Radon-Niodym derivative of the measure P α,m generated by the process Zt with respect to the measure P α, generated by Xt is the following: dp α,m dp α, = exp{ αm ZT + ZT 2 + T 2 α T Ztdt + m 2 + αt 2 T } 2 We can calculate the Radon-Niodym derivative of the shifted process with respect to original one by the help of the Ito's formula from 23 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 { Zt := ZT t/ T : t [, ]} Then Zt = Xt + m/ T, where Xt := XT t/ T, hence d Xt = κ Xt dt + dw t, t R, 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 { Zt : t [, ]} is given by Z + Z + κ Zt dt ã = N a, 2 + κ 2κ + κ 2, 26 see Arató [3] For the variance of ã we have Especially, lim κ κ2 Var ã =, lim Var ã =, lim κ lim κvar ã = κ 2 Var ã =, κ

26 24 CHAPTER 2 PRELIMINARIES hence ã is asymptotically consistent as κ, but Var ã is unbounded as κ Arató in [3] also showed that if the damping parameter and the mean both are unnown in the one-dimensional model, they cannot be estimated well In statistical investigations of independent random variables observations it is well nown that if X, X 2,, X n are normally distributes with parameters m, σ 2, when both of them are unnown, then with an arbitrary degree of condence, a nite condence interval can be constructed, eg, by the help of t-statistics This is not the situation in the case of a stationary Gaussian Marov process In Arató [3] the following is stated When the parameters m and κ of a stationary Gaussian Marov processes are unnown, it is impossible to construct nite condence intervals for m using continuous functionals

27 Chapter 3 Parameter Estimation with Exact Distribution for Multivariate AR Processes 3 MLE of the Multidimensional Case The complex-valued stationary autoregressive process ξt = ξ t + iξ 2 t, t, can be given by the stochastic dierential equation SDE dξt = γξtdt + dwt, where wt = 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 processes and γ = λ iω with λ >, ω R Consider the statistics s 2 ξt = t where θt, t is dened by ξu 2 du, r ξ t = ξt = ξt e iθt t ξu 2 dθu, The process r ξ t = t ξ udξ 2 u ξ 2 udξ u, t, 3 25

28 26 CHAPTER 3 EXACT DISTRIBUTION OF THE MLE is called Levy's stochastic area process It is nown that the maximum lielihood estimate MLE of the period ω is ˆω ξ t = r ξt 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 property This result was rst formulated and applied in astronomy in Arató, Kolmogorov, Sinay [6] Complicated proofs can be found in Noviov [24], Liptser and Shiryayev [2] Recently Arató [4] gave an elegant new proof using Noviov's method The statement can be reformulated also in the following way Let us consider the two-dimensional real-valued stationary autoregressive process Xt, t, given by the SDE dx t dx 2 t λ ω = ω λ X tdt X 2 tdt dw t + dw 2 t, 32 where W t, t, is a standard two-dimensional Wiener process, and λ, ω R Consider the statistics s 2 Xt = t X 2 u + X 2 2 u du, r X t = t The maximum lielihood estimate of the period ω is ˆω X t = r Xt s 2 X t, X u dx 2 u X 2 u dx u and s X t ˆω X t ω D = N, for all t > The following natural question can be formulated process Xt, t, given by the SDE Consider the q-dimensional dxt = AXtdt + 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

29 3 MLE OF THE MULTIDIMENSIONAL CASE 27 have exactly a normal distribution? This process is the so-called Ornstein-Uhlenbec process and considered as a generalization of Wiener processes Pap [25] and Fazeas [4] 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 superuous in [27] Consider the following multidimensional process: m dxt = λi d + ω i C i Xtdt + dw t, X = 33 i= where I d is the d d unit matrix, λ, ω,, ω m C,, C m are xed sew-symmetric matrices R are unown parameters and The MLE of ω = ω,, ω m is given by ˆω X t = σ X tr Xt, where σ X t is the m m matrix t σ X t = C i Xs, C j Xs ds and r X t is the m-dimensional column vector t r X t = C i Xs, dxs In [27] it is proved that i,j m i, m σ /2 X tˆω Xt ω D = N, I m for all t >, 34 if conditions C C3 are satised, where C Ci = 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 LC u C v, u, v m, i, j,, l =,, m where LC 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 superuous Theorem 3 Let Xt, t, be the process given by 33 Let us suppose that the conditions C and C2 satisesd Then 34 holds We need the following lemmas to prove the theorem,

30 28 CHAPTER 3 EXACT DISTRIBUTION OF THE MLE We shall mae use of the following explicite formula which is a special case of Lemma 6 in [2] Lemma 32 Consider a standard d-dimensional Wiener process W t, t For all t let Bt and Qt be d d matrices such that Qt is symmetric, positive semidenite and Then E exp = exp { Tr T T t { 2 Tr T BtB t + Qt dt < 35 t } Bs dw s Qt Bs dw s dt } BtB tγt dt where Γt, t are negative semidenite matrices determined by the Riccati dierential equation Γt = 2Qt ΓtBtBt Γt, ΓT =, Let us denote the cone of the symmetric, positive semidenite d d matrices by C d We shall also use that the distribution of a symmetric, positive semidenite d d random matrix is uniquely determined by the value of its Laplace transform on the cone C d Lemma 33 If σ is a random matrix with σ = σ and σ then the distribution of σ is uniquely determined by the Laplace transform Ψ : C d, given by Ψα := E exp { Trα σ } d d = E exp α ij σ ij, α C d i= j= Proof First we prove that for α C d we have Trα σ It is well nown that there is a symmetric, positive semidenite matrix β C d such that α = β 2 = β β The matrix βσβ is again symmetric and positive denite since βσβ x, x = σβ x, β x, x R d Hence, indeed Trα σ = Trβ βσ = Trβσβ

31 32 PROOF OF THE THEOREM 29 For xed {,, d} let us consider the matrix α C d with entries α ij = Then Trα l σ = σ + 2σ l + σ ll { if i = j =, else Using the classical result on the Laplace transform of a random vector with nonnegative coordinates we now that the joint distribution of the random variables σ : d} {σ + 2σ l + σ ll : l d}, 36 is uniquely determined by the Laplace transform ϕs, d, s l, l d d := E exp s σ = l d s l σ l, s, s l Clearly ϕs, d, s l, l d d = E exp{ s Trα σ Trα l σ} = l d where = E exp{ Trα Σ = Ψα, d α = s α + s l α l C d = l d Consequently the joint distribution of the random variables in 36 is uniquely determined by the Laplace transform Ψ : C d, of the random matrix σ, hence, the joint distribution of the entries of the matrix σ is also uniquely determined by Ψ : C d, since there is a one-to-one correspondence between the entries of σ and the random variables in Proof of the Theorem The proof can be carried out as in [27] We have to show only that for all T > the distribution of the symmetric, positive semidenite random matrix σ X T does not

32 3 CHAPTER 3 EXACT DISTRIBUTION OF THE MLE depend on the parameter ω = ω,, ω m Using Lemma 33 it is sucient to show that the Laplace transform Ψ T α = E exp m i,j= α i,j C i Xt, C j Xt dt, α C d, does not depend on the parameter ω Using the notation we have C := Ψ T α = E exp m i= j= { m α ij Ci C j, T Xt CXtdt Next we show that C is a symmetric, positive semidenite matrix We use again that there exists a matrix β C d such that α = β 2 = β β, hence α ij = d = β iβ j We have 2 m m d d Cx, x = β i βj Ci m C j x, x = β i C i x, thus C C d, indeed Let i= j= = A = λi d + m ω i C i It is nown that the solution Xt, t, of the SDE can be represented in the form Consequently, T Xt CXtdt = T Xt = t t i= = e t sa dw s, i= t e sa dw s e ta Ce ta e sa dw s dt We will show that Lemma 32 can be applied with Bt = e ta and Qt = e ta Ce ta Clearly the condition C and C2 imply BtBt = e 2λt I d }

33 33 SPECIAL CASES 3 and AC = CA, hence Qt = Ce ta e ta = e 2λt C, and we concluded the validity of the condition 35 Applying Lemma 32 and using the above formulae we obtain { } T Ψ T α = exp 2 Tr e 2λt Γtdt, α C d, where Γt, t, is dened by Γt = 2e 2λt C e 2λt Γ 2 t, ΓT = Consequently the Laplace transform Ψ T does not depend on the parameter ω and the proof is completed 33 Special Cases We give some application of the Theorem Corollary Consider the d-dimensional OrnsteinUhlenbec process Xt, t, given by dxt = λi + m ω ic i Xtdt + dw t, X =, i= where C i = C i, i =,, m, C i C j = C j C i, i < j m by where Then the maximum lielihood estimators of the parameters ω,, ω m r i ˆω i X t = ri X t s, i X t2 t t X t = C i Xs, dxs, s i X t = C i Xs, C i Xs ds, are given and s X t ˆω X ω,, s m X t ˆω m X ω m D = Nm, I for all t >

34 32 CHAPTER 3 EXACT DISTRIBUTION OF THE MLE Corollary 2 Consider the d-dimensional process Xt, t, given by dxt = λi + ωcxtdt + dw t, X =, where C = C Then the maximum lielihood estimator of the parameter ω is ˆω X t = r Xt s 2 X t, where and r X t = t CXs, dxs, s 2 Xt = t CXs 2 ds, s X t ˆω X t ω D = N, for all t >

35 Chapter 4 Nearly Unstable AR Models 4 The Shifted AR Process Let m n T,T 2 be the LSE of m n based on the observations {Z n : [T n] [T 2 n]}, where T < T 2 Our purpose is to investigate the limit behaviour of m n T,T 2 as n, and the connection between m n T,T 2 and the MLE m T,T 2 of m based on the observations {Zt : t [T, T 2 ]}, where Zt := Y t + mht is the shifted Ornstein-Uhlenbec process with some appropriate function h We discuss these questions concerning nearly stable, nearly unstable and nearly explosive models, as well It turns out that m n T,T 2 is always asymptotically normal, but the speed of convergence is much less in the nearly unstable case The stationary case will be also treated, when m n T,T 2 might not be asymptotically normal if γ n 4 Case of Zero Start AR Model First, consider a shifted zero start Ornstein-Uhlenbec process {Zt : t } given by dy t = γy t dt + dw t, t, Y =, Zt = Y t + mht, t, where γ R is a nown parameter, h : [, R is a nown function and m R is an unnown parameter 33

36 34 CHAPTER 4 NEARLY UNSTABLE AR MODELS Let T < T 2 Baran, Pap and Zuijlen [7] proved the following result on the MLE of m Denote by P Z and P Y the measures generated on C[T, T 2 ] R by the processes Z and Y, respectively If h is twice continuously dierentiable with h =, then the measures P Z and P Y are equivalent and the MLE of m based on the observations {Zt : t [T, T 2 ]} has the form where γht m T,T 2 = ζ T,T 2 Â T,T 2, γ cothγt ht 2 + γht T2 if γ, T, Â T,T 2 = ht 2 T2 + h t 2 dt, if γ =, T, T T T2 T γ 2 ht 2 + h t 2 dt, γ 2 ht 2 + h t 2 dt, if T =, γ cothγt ht h T ZT + γht 2 + h T 2 ZT 2 T2 + γ 2 ht h t Zt dt, if γ, T, T ht ZT ζ T,T 2 = + h T 2 ZT 2 h T ZT T T 2 T h tzt dt, if γ = T, γht2 + h T 2 T2 ZT 2 + γ 2 ht h t Zt dt, if T = Moreover, m T,T 2 is normally distributed with mean m and variance /ÂT,T 2 We note that the assumption h = is needed because if h and m, then the measures P Z and P Y are singular with respect to each other, since Y = and Z = mh Next, consider a sequence of shifted zero start AR models X n = α n X n + εn, =, 2,, X n =, = X n + m n h n, =,, 2,, Z n

37 4 THE SHIFTED AR PROCESS 35 for n =, 2,, where {α n : n =, 2, } and {h n : n =, 2, ; =,, 2, } are nown and {m n : n =, 2, } are unnown parameters The LSE T,T 2 of the parameter m n based on the observations {Z n : [T n] [T 2 n]}, where T < T 2, can be obtained by minimizing the sum of squares m n 2 Z n [T n] mn h n [T n] Var Z n [T n] + [T 2n] j=[t n]+ if T, and by minimizing the sum of squares 2 Z n j m n h n j α n Z n j mn h n j 4 [T 2n] j= 2 Z n j m n h n j α n Z n j mn h n j 42 if T = We can observe that X n = j= α n j ε n j, hence, Var Z n = Var X n = j= α n 2 j = α n 2 α n 2 if α n, if α n = Consequently, ζ n m n T T,T 2 =,T 2, Â n T,T 2

38 36 CHAPTER 4 NEARLY UNSTABLE AR MODELS where  n T,T 2 = α n 2 2 h n [T n] α n 2[T n] if α n, T, 2 h n [T n] [T n] + [T 2n] j=[t n]+ + h n [T 2n] j=[t n]+ h n j 2, j 2 if α n =, T, ζ n T,T 2 = with the notation [T 2n] h n j 2 if T =, j= α n 2 h n [T n] Zn [T n] α n 2[T n] if α n, T, h n [T n] Zn [T n] [T n] [T 2n] j= + [T 2n] j=[t n]+ + h n j Z n j, if T =, [T 2n] j=[t n]+ h n j Z n j, h n j Z n j, if α n =, T, h n j := h n j α n h n j, Zn j := Z n j α n Z n j We remar that m n T,T 2 has mean m n and variance /Ân T,T 2 and the connec- The next statement describes the asymptotic behaviour of m n tion between discrete and continuous time models Theorem 4 Suppose that h n = h n for n =, 2,, =,, 2, with some twice continuously dierentiable function h : [, R with h = Suppose that for all n, {ε n j : j =,, n} are iid, mean zero and variance

39 4 THE SHIFTED AR PROCESS 37 i If n α n γ R, then m n T,T 2 m n n ii If n α n ±, then D mt,t 2 m D = N n α n m n T,T 2 m n D N,, T2 T  T,T 2 ht 2 dt Proof i First, suppose that T, and for all suciently large n N, α n We investigate the asymptotic behaviour of  n T,T 2 which can be written in the form α  n n T,T 2 = α n 2[T αn α n h n n] [T n] + α n h n [T 2n] [T 2n] + α n 2 j=[t n]+ h n j 2 + α n [T 2n] h n j j=[t n]+ We will prove that nân T,T 2 ÂT,T 2 as n We have Moreover, If γ then n α n γ, α n as n α n 2[Tn] = n n 2[Tn]/n αn e 2γT n n If γ = then Consequently, α n 2 α n = n αn + αn 2[T n] α n 2[T n] α n 2[T n] n α n 2 = n α n 2 n α n 2[T n] [T n] j= { 2γ α n 2j T e 2γT if γ, T if γ = h n j γ e 2γT

40 38 CHAPTER 4 NEARLY UNSTABLE AR MODELS Clearly, we have for all T [T, T 2 ], h n [T n] [T n] = h n Moreover, n n [T 2n] j=[t n]+ [T 2n] h n j j=[t n]+ hence we obtain nân T,T 2 ÂT,T 2 h n j ht 2 T2 ht 2 dt, T T2 h n j 2 h t 2 dt, T Now, we investigate the asymptotic behaviour of ζn T,T 2 which can be written in the form α ζ n T,T 2 = m n  n n 2 T,T 2 + α n 2[T αn α n h n n] [T n] Xn [T n] + α n h n [T 2n] Xn [T 2n] + αn 2 α n [T 2n] h n j+ 2hn j j=[t n]+ [T 2n] j=[t n]+ + h n j Xn j h n j X n j + α n h n [T 2n] hn [T 2n] X n [T 2n] αn h n [T n]+ hn [T n] X n [T n] The random step functions Y n t := n X n [nt], t [T, T 2 ], can be considered as random elements in the Sorohod space D[T, T 2 ] R is nown see, eg, Pap and Zuijlen [26] that It Y n Y, in D[T, T 2 ] R Particularly, for all T [T, T 2 ], X n D n [T n] Y T

41 4 THE SHIFTED AR PROCESS 39 By the Continuous Mapping Theorem we obtain hence, we get [T 2n] n n 3/2 [T 2n] j=[t n]+ h n j+ 2hn j j=[t n]+ ζn n T,T 2 m n  n D T,T 2 Finally, we conclude T2 h n j X n D j hty t dt, T + h n j Xn j T2 D h ty t dt, T γ cothγt ht Y T + γht 2 Y T 2 +h T 2 Y T 2 h T Y T + T 2 T γ 2 ht h t Y t dt, if γ, ht Y T T + h T 2 Y T 2 h T Y T T 2 T h ty t dt, if γ =, = ζ T,T 2 mât,t 2 m n T,T 2 m n n ζ n T,T 2 m n  n T,T 2 = n nân T,T 2 D ζ T,T 2 mât,t 2  T,T 2 = m T,T 2 m D = N,  T,T 2 Next, we suppose that T, and for all suciently large n N, α n = Then, we obviously have nân T,T 2 = Moreover, 2 h n [T n] [T n]/n + n ζ n T,T 2 = m n  n [T 2n] h n j j=[t n]+ T,T 2 + hn [T n] Xn [T n] [T n] h n j 2 ht 2 T2 + h t 2 dt = T ÂT,T 2 T + [T 2n] h n j j=[t n]+ h n j Xn j X n j,

42 4 CHAPTER 4 NEARLY UNSTABLE AR MODELS hence, n ζ n T,T 2 m n  n D ht Y T T,T 2 + h T 2 Y T 2 h T Y T T and we can conclude again that m n T,T 2 m n n T2 = h ty t dt ζ T,T 2 mât,t 2, T D mt,t 2 m D = N The case T = can be handled in a similar way ii Let us start with the case T, Now, we have  T,T 2  n T2 T,T 2 n α n ht 2 dt as n, 2 T since we can use again the representation 43 of α n 2[Tn] = n n 2[Tn]/n αn n Next, we write ζ n T,T 2 ζ n T,T 2 = m n  n T,T 2 + We will prove that ζ n in the form α n 2 h n [T n] Xn [T n] α n 2[T n] T,T 2 m n  n n α n T,T 2 D N,  n T,T 2 { +, + and if n α n,, if n α n + [T 2n] h n j j=[t n]+ T2 T ht 2 dt α n h n j εn j 44 Clearly U n T,T 2 := α n 2 h n [T n] Xn [T n] n α n α n L2, 2[T n]

43 4 THE SHIFTED AR PROCESS 4 since E U n T,T 2 = and Var U n T,T 2 = By the Central Limit Theorem, n α n [T 2n] h n j j=[t n]+ + α n 2 h [Tn] α n 2[T n] n α n α n h n j εn j D N, T2 T ht 2 dt, since Var n α n [T 2n] h n j j=[t n]+ α n h n j εn j = n α n 2 T2 T ht 2 dt [T 2n] h n j j=[t n]+ α n h n j 2 and max Var [T n] j [T 2n] n h j α n h n n α n j εn j h n j α n h n j = max 2 as n [T n] j [T 2n] n α n 2 Consequently, we obtain 44 Hence, nally we obtain = n α n m n T,T 2 m n n α n ζ n T,T 2 m n  n T,T 2 n α n 2 Ân T,T 2 D N, T2 T ht 2 dt The case T = can be handled in a similar way

44 42 CHAPTER 4 NEARLY UNSTABLE AR MODELS Corollary 42 i If α n = γn n m n T,T 2 m n n with γ n γ R, then, D mt,t 2 m D = N Â T,T 2 ii If α n = γn n iii If α n α, then with γ n ±, then n α n m n T,T 2 m n D N n n m T,T 2 m n D N,, T2 T α 2 T 2 T ht 2 dt ht 2 dt Example 43 Let α n = β n q, q R, β, T > and ht = for all t T i If q >, then m n T,T 2 m n n D mt,t 2 m D = N, T, hence the rate of convergence does not depend on q, and the limit law does not depend on β, T 2 ii If q =, then m n T,T 2 m n n D mt,t 2 m = D N, hence the limit law depends on β, T and T 2 iii If q <, then m n T,T 2 m n n q /2 D N, β cothβt + β + β 2, T 2 T β 2, T 2 T hence the rate of convergence depends on q, and the limit law depends on β, T and T 2 Remar 44 We note that no assumption is needed concerning the sequence m n, n =, 2,, unlie in case of the LSE of the coecient α n

45 4 THE SHIFTED AR PROCESS Case of Stationary Nearly Unstable AR Model A stationary AR time series is the wealy stationary solution of where necessarily α < X = α X + ε, Z, Consider a sequence of shifted stationary AR models { Xn Z n n = α Xn + εn = X n, Z, + m n h n, Z, for n =, 2,, where α n <, n =, 2, and h n, n =, 2,, Z are nown and m n, n =, 2, are unnown parameters Suppose again that for all n, {ε n j : j Z} are iid, mean zero and variance Then, E X n =, D 2 Xn = α n 2, for Z The LSE m n of the parameter m n n based on the observations { Z : [T n] [T 2 n]}, where T < T 2, can be obtained by minimizing the sum of squares 4, hence where m n T,T 2 = ζ n T,T 2, Ã n T,T 2 Ã n T,T 2 = α n 2 [T 2 2n] h n [T n] + h n j α n h n j 2, j=[t n]+ ζ n T,T 2 = α n [T2n] 2 h n [T n] Zn [T + n] h n j j=[t n]+ α n h n n n j Z j α Zn j Next, consider a stationary Ornstein-Uhlenbec process {Ỹ t : t R}, which is the stationary solution of where γ > necessarily dỹ t = γỹ t dt + dw t, t R,

46 44 CHAPTER 4 NEARLY UNSTABLE AR MODELS It is a zero mean Gaussian process with EỸ t Ỹ t 2 = e γ t2 t 2γ Consider a shifted stationary Ornstein-Uhlenbec process { Zt : t R}, given by { dỹ t = γỹ t dt + dw t, t R, Zt = Ỹ t + mht, t R, where γ > is a nown parameter, h : R R is a nown function and m R is an unnown parameter In [7] the following result is proved on the MLE of m Denote by P ez and P ey the measures generated on C[T, T 2 ] R by the processes Z and Ỹ, respectively If h is twice continuously dierentiable, then the measures P ez and P ey are equivalent and the MLE of m based on the observations { Zt : t [T, T 2 ]} has the form where m T,T 2 = ζ T,T 2 Ã T,T 2, T2 Ã T,T 2 = γht 2 + γht γ 2 ht 2 + h t 2 dt, T ζ T,T 2 = γht ZT + γht 2 ZT 2 + h T 2 ZT 2 h T ZT T2 + γ 2 ht h t Zt dt T Moreover, m T,T 2 is normally distributed with mean m and variance /ÃT,T 2 Theorem 45 Suppose that twice continuously dierentiable function h : R R Suppose that for all n, {ε n j : j Z} are iid, mean zero and variance h n i If n α n γ, then m n T,T 2 m n n = h n for n =, 2,, Z with some D mt,t 2 m D = N, Ã T,T 2

47 4 THE SHIFTED AR PROCESS 45 ii If n α n then n α n m n T,T 2 m n D N iii Let n α n and ε n N,, T2 a If ht = c for all t R with some c R, then α n m n T,T 2 m n T D N, ht 2 dt 2c 2 b Otherwise, m n T,T 2 m n n D N, T2 T h t 2 dt Proof The proof is similar to the proof of Theorem 4 Example 46 Let α n = β n q, q, β >, and ht = for all t R i If q =, then m n T,T 2 m n n ii If q <, then D mt,t 2 m = D N, m n T,T 2 m n n q /2 D N, hence the rate of convergence depends on q iii If q > and ε n N,, then m n T,T 2 m n n q/2 D N, 2β + β 2 T 2 T β 2, T 2 T, 2β hence the rate of convergence depends on q, but the limit distribution does not depend on T, T 2

48 46 CHAPTER 4 NEARLY UNSTABLE AR MODELS 42 Nearly Unstable Multidimensional AR Processes Our aim was to investigate nearly unstable complexvalued models, where the coef- cient 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, dth component of the process This type of models is rather special However from a practical point of view it is often useful to investigate the behaviour of the greatest eigenvalue We also compared it with the maximum lielihood estimator MLE of the eigenvalue of the coecient matrix of the related continuous time model It is interesting to note that the MLE of the eigenvalue in the related continuous time model depends only on the rst coordinate of the process Matrices consisting of two or more Jordan blocs were also studied 42 Nearly Unstable AR Models with Coecient Matrices in Jordan Normal Form Let C d be the space of the ddimensional 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 Jλ, d := λ λ λ for a d d matrix in Jordan normal form with eigenvalue λ We shall use the short notation Jλ it it does not cause misunderstanding Consider the ddimensional complexvalued autoregressive model { X = JλX + ε, =,2,, X =,

49 42 NEARLY UNSTABLE MULTIDIMENSIONAL AR PROCESSES 47 where λ C is the unnown parameter of the model If we tae into consideration the special form of the coecient matrix then we can calculate the LSE of λ as follows Lemma 42 The LSE of λ, based on the observations X,, X n is given by n d = j= λ := X,j X,j, X,j n d = j= X,j 2, where X = X,,, X,d and X, := Proof Let us consider the sum of squares: Obviously Λ n λ = Λ n λ := n = = n X JλX 2 = n = n = = j= λ X, X,d λ λ λ λ λ X, X, λx, X,2 X, λx,2 X,d X,d λx,d d n X,j X,j 2 λ n = j= = j= 2 d X,j X,j X,j + λ 2 X, X,d d X,j X,j X,j n = j= 2 d X,j 2 Clearly λ is a least squares estimate of λ if Λ n λ is minimal We write the complex number λ in the form λ = α + iβ, where α, β R Rewriting Λ n λ as

50 48 CHAPTER 4 NEARLY UNSTABLE AR MODELS a function of α and β we get the following: Λ n α, β = n = j= d X,j X,j 2 n 2α Re = j= n 2β Im α 2 + β 2 = j= n = j= d X,j X,j X,j + d X,j X,j X,j + d X,j 2 This expression taes minimum if the following equations hold, 2 2 n = j= n = j= d Re d Im [ ] X,j X,j X,j + 2α [ ] X,j X,j X,j + 2β n = j= n = j= Hence the LSE of α and β are given by n [ ] d = j= Re X,j X,j X,j α n = n d = j= X,,j 2 d X,j 2 =, d X,j 2 = n [ ] d = j= β Im X,j X,j X,j n = n d = j= X,j 2 Thus we obtain the assertion of the lemma 422 Convergence of a Related Step Process For n =, 2, consider the ddimensional complexvalued AR model: { X n = Jλ n X n + εn, =, 2,, n, X n =, 45

51 42 NEARLY UNSTABLE MULTIDIMENSIONAL AR PROCESSES 49 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 Jλ n Je iθ since λ n e iθ It is easy to see that the model is nearly unstable, since ϱje iθ = We consider the following transformation on the elements of C d : z e iθ z This transformation can be considered as a rotation in C d, since e iθ u, e iθ v = u, v for u, v C d We dene the following rotated elements the construction is the same as in Pap, Zuijlen [26]: Z n ζ n := e iθ X n, := e iθ ε n Let Jλ n := e iθ Jλ n = e hn/n e iθ e iθ e hn/n The following model is clearly equivalent with 45: { Z n = Jλ n Z n + ζn, =, 2,, n, Z n = Now we consider the normalization matrices { L n := diag n /2, n 3/2 e iθ,, n d+/2 e d iθ}, and random step functions Y n t := e i[nt]θ L n X n [nt] = L nz n [nt], M n t := [nt] n = e iθ ε n = [nt] n = ζ n Let us put the following condition on the random disturbances {ε n }