Let there be q quarterly observable variables and m

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Let there be Q quarterly observable variables and M monthly observable variables, and let us summarize them by vector y t = [ y q 1 ,t , ..., y q Q,t , y m Q +1 ,t , ..., y m Q + M,t ] 0 . To summarize equations ( A.3 )–( A.6 ) into an observation equation, we need to define the vector of unobserved states z t = [ f t , ..., f t - 4 , u 1 ,t , ..., u 1 ,t - 4 , ..., u Q,t , ..., u Q,t - 4 , u Q +1 ,t , ..., u Q +1 ,t - P +1 , ..., u Q + M,t , ..., u Q + M,t - P +1 ] 0 . As we can see, the vector z t combines the common factor with lags up to 4 and individual compo- nents of the quarterly variables with lags up to 4 in order to account for the representation of quarterly variables according to equation ( A.6 ); it also includes individual components of the monthly variables with lags up to P - 1. 14 Then, assuming that all the variables are observed in period t , we can formulate the observation equation: y t = H z t + η t , η t ∼ N (0 , R ) (A.7) where η t is a vector of measurement errors, and ( Q + M ) × (5 + 5 Q + PM ) matrix H is the following: H = γ 1 × H Q H Q · · · 0 0 · · · 0 . . . . . . . . . . . . . . . . . . . . . γ Q × H Q 0 · · · H Q 0 · · · 0 γ Q +1 × H 5 M 0 · · · 0 H P M · · · 0 . . . . . . . . . . . . . . . . . . . . . γ Q + M × H 5 M 0 · · · 0 0 · · · H P M , (A.8) H Q = h 1 3 2 3 1 2 3 1 3 i , H 5 M = h 1 0 0 0 0 i , H P M = h 1 0 ... 0 i . 14 Because the lags for quarterly variables’ individual components are capped at 4, this specification effectively restricts P to be no greater then 5. This restriction can easily be relaxed. ECB Working Paper Series No 2381 / March 2020 38
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Note that the size of the matrix H P M is 1 × P , and the only non-zero element is the first one. More generally, in periods when some of the observations are missing, the observation equation can be cast without the rows that correspond to the missing observations: y * t = H t z t + η * t , η * t ∼ N (0 , R t ) (A.9) where H t is obtained by taking H and eliminating the columns that correspond to the missing variables, and the matrix R t is obtained by eliminating the corresponding rows and columns from matrix R . Next, let us define the dynamics of the unobserved state z t : z t = s t μ 0 + (1 - s t ) μ 1 + s t x t 0 . . . 0 + F z t - 1 + ε t , ε t ∼ N (0 , Q ) , (A.10) In this equation, F is a (5 + 5 Q + PM ) × (5 + 5 Q + PM ) matrix, which can be compactly expressed as follows: F = F 0 0 · · · 0 0 Ψ 1 · · · 0 . . . . . . . . . . . . 0 0 · · · Ψ M + Q , where F 0 = 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 . As for the matrix Ψ i , it is a 5 × 5 matrix for quarterly series ( i = 1 , ..., Q ), since there are four lags of monthly individual components for each quarterly series in the state vector z t : Ψ i = ψ i, 1 ψ i, 2 ψ i, 3 ψ i, 4 ψ i, 5 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 ECB Working Paper Series No 2381 / March 2020 39
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However, the coefficients ψ i,p are non-zero only for p < P , where P is the number of specified lags. 15 In case of monthly series ( i = Q + 1 , ..., Q + M ), the matrix Ψ i is P × P . Vector ε t contains shocks to the common factor and each individual component: ε t = h [ e f,t , 0 , 0 , 0 , 0] , [ e 1 ,t , 0 , 0 , 0 , 0] , ..., [ e M + Q,t , 0 , 0 , 0 , 0] i 0 . (A.11) Correspondingly, the matrix Q is a diagonal matrix, such that diag ( Q ) = [ σ 2 f , 0 , 0 , 0 , 0] , [ σ 2 1 , 0 , 0 , 0 , 0] , ..., [ σ 2 M + Q , 0 , 0 , 0 , 0] . (A.12) Note that the dynamics equation ( A.10 ) contains an extra vector that depends on the state indicator s t and the latent variable x t
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  • Economics, Recession, Late-2000s recession, GWI

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