Career

A Test for Comparing Two Discrete Stochastic Dynamical Systems Under Heteroskedasticity

Description
A Test for Comparing Two Discrete Stochastic Dynamical Systems Under Heteroskedasticity
Categories
Published
of 26
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Transcript
  Differ Equ Dyn Syst (July 2011) 19(3):211–236DOI 10.1007/s12591-011-0085-3ORIGINAL RESEARCH A Test for Comparing Two Discrete Stochastic DynamicalSystems Under Heteroskedasticity Gladys E. Salcedo  ·  Pedro A. Morettin  · Clélia M. C. Toloi Published online: 20 April 2011© Foundation for Scientific Research and Technological Innovation 2011 Abstract  Inthispaperwedevelopastatistictestinordertocomparetwodiscretestochasticdynamicalsystemswithautoregressivestructuresunderheteroskedasticity,morespecificallythe variance of the errors follows a general function depending on time. The random distur-bances or errors can also be contemporaneously correlated through some structure varyingin time. The approach is based on a Wald test and we present three consistent estimates forthe asymptotic covariance matrix. The study of the asymptotic properties is based on the  L  p -mixingale processes theory since we consider errors generated from martingale differ-enceprocesses.Inordertoevaluatetheperformanceofthetestinfinitesamples,weransomesimulations with satisfactory results. A real application is also provided. Keywords  AR models  ·  Hypothesis testing  ·  Heterogeneity  ·  L  p -mixingales  ·  Wavelets Introduction In recent years there has been increasing interest in the application of models in time seriesanalysis. This is due to the development of a wide variety of models for time series analy-sis and their adaptability to a wide range of applications, including molecular biology andgenetics.A time series can be seen as the output of a dynamic system perturbed by random distur-bances(seee.g.[16,21,22]).Atimeseriesisinterpretednaturallyasacombinationofseveral G. E. Salcedo ( B )Department of Mathematics, University of Quindío, Bolivar Ave., AA 460 Armenia, Colombiae-mail: gsalcedoe@gmail.comP. A. Morettin  ·  C. M. C. ToloiInstitute of Mathematics and Statistics, University of São Paulo, Rua do Matão 1010, São Paulo,SP 05508-090, Brazile-mail: pam@ime.usp.brC. M. C. Toloie-mail: clelia@ime.usp.br  1 3  212 Differ Equ Dyn Syst (July 2011) 19(3):211–236 components as trend, seasonal and autoregressive components. Among these, Autoregres-sive (AR) models form a very important class of discrete dynamical models for stationarystochastic processes, due to the fact that they can model a wide variety of phenomena, theyare easy to estimate and interpret and the asymptotic properties of autoregressive estimatorsare well understood. Due to these advantages, autoregressive models have been extended toincludedifferentkindsofstochasticprocesses,fromstationarytolocallystationaryprocesses[8], including discrete and continuous systems [4,15]. Autoregressive models where the non stationarity is due to time-dependent parametershave been considered by Dahlhaus [8]. An extension to linear stochastic systems was givenby Chiann and Morettin [6] and for vectorial systems by Sato et al. [25]. In the case of Auto- regressive Moving Average (ARMA) models, Basawa and Lund [1] considered periodicallyvarying coefficients and their models were extended recently by Francq and Gautier [12] to nonperiodictime-varyingcoefficientswhicharesubjecttoirregularregimechangesatknowntime points. Phillips and Xu [23] developed some procedures for making robust inferencesfor general forms of heteroskedasticity in the context of a finite-order stable autoregression.Comparison of two discrete stochastic systems has been a problem of interest for manyauthors. Most existing comparison techniques are applicable to systems that are stationary,or to non stationary ones that can be transformed to stationary by some simple transforma-tions such as differencing (see e.g. [2,11,13,17,18,26]). For non stationary systems that can not be transformed into stationary ones, Maharaj [19,20] developed some methods using evolutionary spectral approaches, and Salcedo et al. [24] developed methods based on theautoregressive parameters varying in time with Gaussian disturbances.In this paper we develop a robust test in order to compare the parameters of two dis-crete stochastic systems with an autoregressive representation and under heteroskedasticity.More specifically, the variance of the disturbances follows a general function depending ontime. The random disturbances of the two systems can also be contemporaneously correlatedthrough some structure varying in time. We also present three consistent estimates for theasymptotic covariance matrix based on the ordinary least squares residuals, similar to thoseprovided in [23], but in one of them, we consider a wavelet expansion for the estimation of the variance function. The study of the asymptotic properties is based on the  L  p -mixingaleprocesses theory [10] as the errors are generated from martingale difference processes. Toevaluate the performance of the test in finite samples, we present some simulation results.The plan of the remainder of the paper is as follows. Section 2 presents the model andassumptions. In Sect. 3 we present the hypotheses and develop the limit theory of the het-eroskedasticity-robust test for comparing the two vectors of autoregressive parameters. Theinference is given in Sect. 4. Some simulation results on the size and power of the test for finite samples are given in Sect. 5. A real application is provided in Sect. 6. In Sect. 7, the conclusions are discussed. Model and Assumptions Let us consider two discrete dynamical systems with the autoregressive representation of order  p ,  AR (  p ) :  X  t   =  θ  0  +  θ  1  X  t  − 1  + ··· +  θ   p  X  t  −  p  +  ε  x  , t  , and Y  t   =  φ 0  +  φ 1 Y  t  − 1  + ··· +  φ  p Y  t  −  p  +  ε  y , t  ,  1 3  Differ Equ Dyn Syst (July 2011) 19(3):211–236 213 where the lag order  p  is finite and known,  θ   p  =  0 and  φ  p  =  0. The disturbances  ε  x  , t   and ε  y , t   are distributed according to  ε  x  , t   =  σ   x  (  t T   ) u t   and  ε  y , t   =  σ   y (  t T   )v t  , and they can be con-temporaneously correlated through a function  σ   xy (  t T   ) . We state the following assumptions:(A1) All roots of the characteristic polynomials 1  −  θ  1  z  −  θ  2  z 2 − ··· −  θ   p  z  p =  0 and1  −  φ 1  z  −  φ 2  z 2 − ··· −  φ  p  z  p =  0 lie outside the unit circle.(A2)  σ   x  ( r  ) , σ   y ( r  ) , and  σ   xy ( r  ) are non-stochastic, positive, measurable, uniformly boundedon the interval  ( −∞ , 1 ] , with a finite number of points of discontinuity, and satisfy aLipschitz condition, except at points of discontinuity.(A3)  { u t  }  and  { v t  }  are strong mixing ( α -mixing) martingale difference processes with  E  { u t  | F  t  − 1 } =  E  { v t  | F  t  − 1 } =  0 ,  E  { u 2 t   | F  t  − 1 } =  E  { v 2 t   | F  t  − 1 } =  1 ,  a . s . for all  t  , and contemporaneously correlated through the function  E  { u t  v t  | F  t  − 1 } = ρ(  t T   ),  a . s .  for all  t  , with the natural filtration F  t   =  σ(( u s ,v s ), s  ≤  t  )  and  ρ   t T    = σ   xy  t T   σ   x   t T   σ   y  t T   . (A4) There exist a  δ >  1 and constants  C  1 , C  2  >  0, such that sup t   E  ( u 4 δ t   ) <  C  1  <  ∞ , andsup t   E  (v 4 δ t   ) <  C  2  <  ∞ .  Remarks  1. Under assumption (A1), for the stochastic process  {  X  t  , t   ∈ Z  }  we have: •  The autoregressive coefficients satisfy the usual stability conditions which, if   σ   x  ( · ) is a constant function and { ε  x  , t  } homoskedastic, {  X  t  } will be stationary or asymptoti-callycovariance-stationary,dependingoninitialconditions.Inthiscase, µ  x   =  E  (  X  t  ) exists and is given by  µ  x   =  θ  0 1 − θ  1 −···− θ   p . • {  X  t  }  has a Wold representation given by  X  t   =  µ  x   +  ∞ i = 0  α i ε  x  , t  − i , in which thesequence  { α i }  satisfies  ∞ i = 0  | α i |  <  ∞  and α i  −  θ  1 α i − 1  − ··· −  θ   p α i −  p  =  0 for  i  >  0 .α 0  =  1 and  α i  =  0 for  i  <  0 . Define    x   to be the  p  ×  p  matrix with  ij th element given by γ  | i −  j |  =  γ   x  ( k  )  = ∞  i = 0 α i α i + k   <  ∞ ,  k   = | i  −  j | =  0 , 1 ,...,  p  −  1 , where the inequality follows from  |  ∞ i = 0  α i α i + k  | ≤  ∞ i = 0  | α i | 2 <  ∞ . In the ho-moskedastic case, when  σ   x  ( · )  =  σ   x  ,  σ  2  x  γ   x  ( k  ) , for  k   =  0 , 1 , 2 ,... , is the covariancesequence of the covariance-stationary process  {  X  t  , t   ∈ Z  } , •  The same remarks are valid for the process  { Y  t  , t   ∈  Z  }  for which the Wold repre-sentation is given by  Y  t   =  µ  y  +  ∞ i = 0  δ i ε  y , t  − i , were  { δ i } ,  µ  y  and    y  are similarlydefined. •  If  γ   xy (τ)  =  γ   xy ( l  −  h )  = ∞  i = 0 α i δ i + l − h  = ∞  i = 0 α i δ i + τ   <  ∞ , we can also define the  p  ×  p  matrix    xy  with  hl th element given by  γ   xy ( l  −  h ) , h , l  =  1 ,...,  p . Thus, if   σ   xy ( · )  =  σ   xy , the function  σ   xy γ   xy ( l  −  h ) , represents thecross-covariance structure of the processes in the stationary case.  1 3  214 Differ Equ Dyn Syst (July 2011) 19(3):211–236 2. Under assumption (A2), the functions  σ   x  ( · ),σ   y ( · )  and  σ   xy ( · )  are integrable on  [ 0 , 1 ] , upto any finite order. To simplify the notation, we write   10  σ  m ( r  ) dr   =    σ  m .3. Byassumption(A3),  E  ( u s u t  )  =  E  (v s v t  )  =  E  ( u s v t  )  =  0for s  =  t  ,andtheassumptions  E  { u 2 t   | F  t  − 1 } =  E  { v 2 t   | F  t  − 1 } =  1 and  E  { u t  v t  | F  t  − 1 } =  ρ(  t T   )  are used for normalization,so that the variance and covariance functions are identified, i.e,  E  (ε 2  x  , t  )  =  E    E  (ε 2  x  , t  | F  t  − 1 )   =  σ  2  x   ( t  / T  ),  E  (ε 2  y , t  )  =  E    E  (ε 2  y , t  | F  t  − 1 )   =  σ  2  y  ( t  / T  ) and  E  (ε  x  , t  ε  y , t  )  =  E    E  (ε  x  , t  ε  y , t  | F  t  − 1 )   =  σ   xy  ( t  / T  ). 4. By assumption (A4) and Liapunov’s inequality,  E  | u t  | η and  E  | v t  | η exist for all  η  ≤  4 δ and all expectations involving up to any four combinations of them. Testing Hypotheses and Limit Theory Suppose that we have  T   +  p  observations from each process  {  X  t  , t   ∈  Z  }  and  { Y  t  , t   ∈  Z  } denoted by  X  −  p + 1 ,  X  −  p + 2 ,...,  X  0 ,  X  1 ,...,  X  T   and  Y  −  p + 1 , Y  −  p + 2 ,..., Y  0 , Y  1 ,..., Y  T  , respectively. The last  T   observations  X  1 ,...,  X  T   can be represented in matrix form as x  =  A  x    + ε  x  ,  (1)with  x  =  (  X  1 ,  X  2 ,...,  X  T  ) ′ ,    =  (θ  0 ,θ  1 ,...,θ   p ) ′ ,  ε  x   =  (ε  x  , 1 ,ε  x  , 2 ,...,ε  x  , T  ) ′ , and  A  x   =  1  X  0  X  − 1  ...  X  −  p + 1 1  X  1  X  0  ...  X  −  p + 2 ......... 1  X  T  − 1  X  T  − 2  ...  X  T  −  p  T  × (  p + 1 ) . Similarly, the observations  Y  1 ,..., Y  T   can be represented by y  =  A  y   + ε  y ,  (2)with  y ,  A  y ,   , and  ε  y  analogously defined.We want to decide if the two series  {  X  t  , t   =  1 ,..., T  }  and  { Y  t  , t   =  1 ,..., T  }  weregenerated by the same heteroskedastic autoregressive system. Thus, according to (1) and (2) the problem is reduced to test the hypotheses  H  0  :    =  ,  (3)  H  1  :    =  . To define the test statistic, we consider the joint model, Z  =  X β  + ε ,  (4)where Z  =  xy  2 T  × 1 ,  X  =   A  x   00  A  y  2 T  × 2 (  p + 1 ) ,  β  =    2 (  p + 1 ) × 1 and  ε  =  ε  x  ε  y  2 T  × 1 .  1 3  Differ Equ Dyn Syst (July 2011) 19(3):211–236 215 Notice that, under model (4), the hypotheses defined in (3) are equivalent to  H  0  :  C β  =  0  (5)  H  1  :  C β  =  0 , with  C  = [  I   p + 1  −  I   p + 1 ] , where  I   p + 1  is the  (  p  +  1 )  ×  (  p  +  1 )  identity matrix.For the model (4), the ordinary least squares (OLS) estimator of   β  is given by  ˆ β  = ( X ′ X ) − 1 X ′ Z  and, under assumptions (A1)–(A4) from Sect. 2, we present some statistical properties that allow us to obtain the statistic for testing (5) and its asymptotic distribution. Lemma 1  Let l  p  be the column p-vector   ( 1 , 1 ,..., 1 ) ′ . Under the stated assumptions, asT   → ∞  , (i)  1 T   X ′ ε  p −→  0 . (ii)  1 T   X ′ X  p −→   1  =    xx   00    yy  , with   xx   =   1  µ  x  l ′  p µ  x  l  p  µ 2  x   +    σ  2  x     x   ,   yy  =   1  µ  y l ′  p µ  y l  p  µ 2  y  +    σ  2  y    y  , where  µ  x   and     x   ( µ  y  and     y  , respectively) are defined in Remark 1 from Sect. 2.Proof   Note that  1 T   X ′ ε  =  1 T   T t  = 1 e t  , where  e ′ t   =  (ε  x  , t  ,  X  t  − 1 ε  x  , t  ,...,  X  t  −  p ε  x  , t  ,ε  y , t  , Y  t  − 1 ε  y , t  ,..., Y  t  −  p ε  y , t  ) . Thus, to prove (i) it suffices to show that: •  1 T   T t  = 1  ε  x  , t  p −→  0, and similarly,  1 T   T t  = 1  ε  y , t  p −→  0. •  1 T   T t  = 1  X  t  − r  ε  x  , t  p −→  0 ,  ∀ r   =  1 , 2 ,...,  p , and similarly,  1 T   T t  = 1  Y  t  − r  ε  y , t  p −→  0.The proofs are given below and then item (i) follows since each component of   1 T   X ′ ε converges in probability to zero. •  Notice that  { ε  x  , t  , F  t  }  is a martingale sequence since  E  (ε  x  , t  | F  t  − 1 )  =  E   σ   x    t T   u t  | F  t  − 1   =  σ   x    t T    E  ( u t  | F  t  − 1 )  =  0 . On the other hand,  E  (ε 2  x  , t  )  =  σ  2  x  (  t T   ) <  ∞ , by assumption (A3). By the law of largenumbers (LLN) for martingale difference sequences (  L 1 -mixingales),1 T  T   t  = 1 ε  x  , t  p −→  0 . •  Similarly,  {  X  t  − r  ε  x  , t  , F  t  }  is a martingale difference sequence, since  E  (  X  t  − r  ε  x  , t  | F  t  − 1 )  =  E   µ  x   + ∞  i = 0 α i ε  x  , t  − r  − i  ε  x  , t   | F  t  − 1   =  X  t  − r   E  (ε  x  , t  | F  t  − 1 )  =  0 . On the other hand,  E  (  X  2 t  − r  ε 2  x  , t  )  ≤    E  (  X  4 t  − r  )  E  (ε 4  x  , t  )  ≤   sup t   E  (  X  4 t  − r  ) sup t   E  (ε 4  x  , t  ) <  ∞ ,  1 3
Search
Similar documents
View more...
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks