Topics in Structural VAR Econometrics by Prof. Carlo Giannini (auth.)

By Prof. Carlo Giannini (auth.)

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In practical applications condition a) can be used and checked numerically remembering touse vecC =Sc 'Yc + Sc and assigning "random" numbers to the elements ofthe 'Yc vector in order to insert a "proper'' matrix in the (/®C) nucleus of the fonnula. The numerical check of the condition does not contribute much to understanding the role and working of different typical constraints. In appendix C, using condition b), we will propose a symbolic analysis of some interesting cases. L. estimation. Still trying to avoid using Lagrange multipliers technicp1e, we will use the following restrictions expressed in explicit fonn vecC =Sc 'Yc + Sc 1 Obviously, the "true" vector vec(Co) must satisfy the consttaints Re vec C =dc in implicit form or vec C =Sc "(c + Sc in explicit fonn.

29 using the chain rule of differentiation we can find the score vector for the vector of the "free" elements "(c: ! '( c) = "(; a L avecK . avecC = avecK avecC aYc =- f' (vec K) ( C' - 1®C"" 1)sc = f' (vec C) Sc The first order condition for the maximization of the log-likelihood with respect to "(c are: f'(Yc) =f' (vec C) Sc= [0] 1xl in row form, or f("(c) =Sc' f(vec C) = [0] lx1 in column form. Taking into account that J Ir(Yc) =E [ f(yc) · f'(yc) =Sc' [f(vecC) ·f'(vecC)] Sc, weobtain IT(yc) =Sc' [Ir vecCJSc and obviously I(yc) = plim r->oo ~Ir (Yc) =Sc' [/ vecC] Sc.

Pi=lAiJ'Po or recursive1y } Pi=Ci·Po '-"" C·- ~ C· ·A· i= 1, ... and Co=In j=l ,_, '} In order to arrive at the asymptotic distribution of the estimated Pi we use the following additional notation: 1t = 2 vecTI = vec [Al, Az, ... _h and d fTO k=O The l:(h)ii (n2xn2) block is the variance covariance matrix of the Pi (n2xl) vector of "structuralized" impulse responses, where Pi= vecPi.

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