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- ξRnR                      where F is a (T à 1) global factor vector and Dc are block factor vectors. Following Doz et al. (2012) we compute the log-likelihood, l(Y, Z, θ), associated to the system composed by the equations (12) and (13) as a function of the data matrix Y , the set of parameters θ = {Î, R, A, Q} and the matrix of factors Z(see appendix B of BaÅbura and Modugno, 2014, for the explicit form.) Then, we proceed to the maximization of l through the following two steps procedure (which details are exposed below): 1. Given the factors, the parameters are derived by analytical maximization of l. 2. Given the parameters, the factors are derived running Kalman Filter/Smoother on the system of equations (12) and (13). This procedure defines a sequence of increasing log-likelihood values l Y, F(0) , θ(0) â l Y, F(0) , θ(1)
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- X t=1 Pâ1,g,t !0 (25) (26) Estimates of the complete form parameters AÌ(k+1) , QÌ(k+1) are obtained by trivial recomposition of the block structure. The updated estimates for the loadings matrix can be seen as the NA-corrected OLS solutions of the regressions (see BaÅbura and Modugno, 2014, pag. 138, eq. (11))17 yt = ÎgZg,t + ÎcZc,t + vt c = 1, . . . , C (27) for nb series of the block b. vec(Îb) = T X t=1 Zgc,tZ gc,t â IndNA t !â1 vec T
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