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By Prof. Carlo Giannini (auth.)

ISBN-10: 3540552626

ISBN-13: 9783540552628

ISBN-10: 3662027577

ISBN-13: 9783662027578

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2. L [0] 2 _jQJ_ [0] 2 lra [0] and postmultiply by the block-diagonal invertible matrix I [0] ] [ K'®B [0] -(I®B) ' arriving at the equivalent (ie, same rank) matrix v* Nn Nn ] vecA = [ , Nn N n [vecB] Ra(K ®B) [0] · [0] Rb(l®B) From the property ofthe Nn matrix (see Magnus (1988) p. 49, property IV) Nn (A®A) =Nn(A®A)Nn =(A®A)Nn if we postmultiply v* [ [~:~~] by the following matrix (B'®B') [0] ] [0] (B'®B') 38 by virtue of the fact that I Nn][ (B'®B') [0] ]-[ (B'®B') [0] [ Nn Nn Nn [0] (B'®B') [0] (B'®B') ][NnNn INnNn] ' premultiplying by the block-diagonal invertible matrix B,-I®B,-1 [0] [0] [0] [0] Br-I®B,-1 [0] [0] we arrive at the following equivalent matrix: v* * Nn Nn ] vecA Nn Nn [ [vecB] = Ra(A'®BB') [0] .

4. L. Estimation for the AB-Model The AB-model is completely defmed by the following equations and distributional assumptions A et=Bet A and B invertible matrices of order n E(et) = [0] E(ete'r) =I er- I MN ([0], :E) det :E *0 1\ All the sample information is contained in the :E matrix The corresponding log-likelihood function for the parameters of interest in the AB-model (the 2n2 parameters in theA and B matrices) is L(A,B) = c + TiogiAI- TiogiBI- ~tr(A'B' -lB-1A ~) (1) We can write this log-likelihood also in the following form T L (A,B) = c + TiogiKI- -ztr(K' K ":E) (lb) withK=B-1 A.

G:] be non zero at ßfor ,m. ff[g{ß) - g(p)] _! N [ [0], g ff ff] l: ( J (where ~ is a mxn matrix and (~ ~ is a 10<111 ma1rix). On the basis of this theorem we can calculate the distribution of vecK* where K* = ß-1 A, and K = B-1 A. Starting from the distribution of[~:~;] and remembering that . iJvecK =[I®B-11-(A'B,-1®B-1) () vecA vecB 3 J Our formulation is substantially identical to Lütkepohl's (1990), minor changes are due to our modified differential notation. For a more rigorous treaunent of the problern faced here andin the following pages see Serfling (1980) p.

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

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