TimeSeriesBook.pdf

Appendix autoregressive final form definition 121

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Appendix: Autoregressive Final Form Definition 12.1 defined the VARMA process { X t } as a solution to the cor- responding multivariate stochastic difference equation (12.1). However, as pointed out by Zellner and Palm (1974) there is an equivalent representation in the form of n univariate ARMA processes, one for each X it . Formally, these representations, also called autoregressive final form or transfer func- tion form (Box and Jenkins, 1976), can be written as det Φ(L) X it = [Φ * (L)Θ(L)] i Z t
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240 CHAPTER 12. VARMA PROCESSES where the index i indicates the i -th row of Φ * (L)Θ(L). Thereby Φ * (L) denotes the adjugate matrix of Φ(L). 5 Thus each variable in X t may be investigated separately as an univariate ARMA process. Thereby the au- toregressive part will be the same for each variable. Note, however, that the moving-average processes will be correlated across variables. The disadvantage of this approach is that it involves rather long AR and MA lags as will become clear from the following example. 6 Take a simple two-dimensional VAR of order one, i.e. X t = Φ X t - 1 + Z t , Z t WN(0 , Σ). Then the implied univariate processes will be ARMA(2,1) processes. After some straightforward manipulations we obtain: (1 - ( φ 11 + φ 22 )L + ( φ 11 φ 22 - φ 12 φ 21 )L 2 ) X 1 t = Z 1 t - φ 22 Z 1 ,t - 1 + φ 12 Z 2 ,t - 1 , (1 - ( φ 11 + φ 22 )L + ( φ 11 φ 22 - φ 12 φ 21 )L 2 ) X 2 t = φ 21 Z 1 ,t - 1 + Z 2 t - φ 11 Z 2 ,t - 1 . It can be shown by the means given in Sections 1.4.3 and 1.5.1 that the right hand sides are observationally equivalent to MA(1) processes. 5 The elements of the adjugate matrix A * of some matrix A are given by [ A * ] ij = ( - 1) i + j M ij where M ij is the minor (minor determinant) obtained by deleting the i -th column and the j -th row of A (Meyer, 2000, p. 477). 6 The degrees of the AR and the MA polynomial can be as large as np and ( n - 1) p + q , respectively.
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Chapter 13 Estimation of Vector Autoregressive Models 13.1 Introduction In this chapter we derive the Least-Squares (LS) estimator for vectorautore- gressive (VAR) models and its asymptotic distribution. For this end, we have to make several assumption which we maintain throughout this chapter. Assumption 13.1. The VAR process { X t } is generated by Φ(L) X t = Z t X t - Φ 1 X t - 1 - · · · - Φ p X t - p = Z t with Z t WN(0 , Σ) , Σ nonsingular, and admits a stationary and causal representation with respect to { Z t } : X t = Z t + Ψ 1 Z t - 1 + Ψ 2 Z t - 2 + . . . = X j =0 Ψ j Z t = Ψ(L) Z t with j =0 k Ψ j k < . Assumption 13.2. The residual process { Z t } is not only white noise, but also independently and identically distributed: Z t IID(0 , Σ) . Assumption 13.3. All fourth moments of Z t exist. In particular, there exists a finite constant c > 0 such that E ( Z it Z jt Z kt Z lt ) c for all i, j, k, l = 1 , 2 , . . . , n, and for all t. 241
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242 CHAPTER 13. ESTIMATION OF VAR MODELS Note that the moment condition is automatically fulfilled by Gaussian processes. For the ease of exposition, we omit a constant in the VAR. Thus, we consider the demeaned process.
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