TimeSeriesBook.pdf

Exercise 1756 consider the state space model of an

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Exercise 17.5.6. Consider the state space model of an AR(1) process with measurement error analyzed in Section 17.2 : X t +1 = φX t + v t +1 , v t IIDN(0 , σ 2 v ) Y t = X t + w t , w t IIDN(0 , σ 2 w ) . For simplicity assume that | φ | < 1 . (i) Show that { Y t } is an ARMA(1,1) process given by Y t - φY t - 1 = Z t + θZ t - 1 with Z t WN(0 , σ 2 Z ) .
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366 CHAPTER 17. KALMAN FILTER (ii) Show that the parameters of the state space, φ, σ 2 v , σ 2 w and those of the ARMA(1,1) model are related by the equation θσ 2 Z = - φσ 2 w 1 1 + θ 2 = - φσ 2 w σ 2 v + (1 + φ 2 ) σ 2 w (iii) Why is there an identification problem?
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Appendix A Complex Numbers The simple quadratic equation x 2 + 1 = 0 has no solution in the field of real numbers, R . Thus, it is necessary to envisage the larger field of complex numbers C . A complex number z is an ordered pair ( a, b ) of real numbers where ordered means that we regard ( a, b ) and ( b, a ) as distinct if a 6 = b . Let x = ( a, b ) and y = ( c, d ) be two complex numbers. Then we endow the set of complex numbers with an addition and a multiplication in the following way: addition: x + y = ( a, b ) + ( c, d ) = ( a + c, b + d ) multiplication: xy = ( a, b )( c, d ) = ( ac - bd, ad + bc ) . These two operations will turn C into a field where (0 , 0) and (1 , 0) play the role of 0 and 1. 1 The real numbers R are embedded into C because we identify any a R with ( a, 0) C . The number ı = (0 , 1) is of special interest. It solves the equation x 2 +1 = 0, i.e. ı 2 = - 1. The other solution being - ı = (0 , - 1). Thus any complex number ( a, b ) may be written as ( a, b ) = a + ıb where a, b are arbitrary real numbers. 2 1 Substraction and division can be defined accordingly: subtraction: ( a, b ) - ( c, d ) = ( a - c, b - d ) division: ( a, b ) / ( c, d ) = ( ac + bd, bc - ad ) ( c 2 + d 2 ) , c 2 + d 2 6 = 0 . 2 A more detailed introduction of complex numbers can be found in Rudin (1976) or any other mathematics textbook. 367
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368 APPENDIX A. COMPLEX NUMBERS -2 -1 0 1 2 -2 -1 0 1 2 real part imaginary part unit circle: a 2 + b 2 = 1 a b z = a + ib r θ -b conjugate of z: a - ib Figure A.1: Representation of a complex number An element z in this field can be represented in two ways: z = a + ıb Cartesian coordinates = re ıθ = r (cos θ + ı sin θ ) polar coordinates . In the representation in Cartesian coordinates a = Re( z ) = < ( z ) is called the real part whereas b = Im( z ) = = ( z ) is called the imaginary part of z . A complex number z can be viewed as a point in the two-dimensional Cartesian coordinate system with coordinates ( a, b ). This geometric inter- pretation is represented in Figure A.1. The absolute value or modulus of z , denoted by | z | , is given by r = a 2 + b 2 . Thus, the absolute value is nothing but the distance of z viewed as a point in the complex plane (the two-dimensional Cartesian coordinate system) to the origin (see Figure A.1). θ denotes the angle to the positive real axis (x-axis) measured in radians. It is denoted by θ = arg z . It holds that tan θ = b a . Finally, the conjugate of z , denoted by ¯ z , is defined by ¯ z = a - ıb .
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