C the prior distribution for μ and μ1 is normal μ

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c. The prior distribution for μ 0 and μ 1 is normal: μ 0 μ 1 ∼ N ( a, V ) . Define Y * = { y * t } , such that y * t = f i t - s i t × x i t , and X * = [ 1 - S i , S i ]. Then, the posterior distribution is also normal, N a, ¯ V ), such that ¯ V = V - 1 + ( X * ) 0 X * - 1 ; ¯ a = ¯ V V - 1 a + ( X * ) 0 Y * . The regime-specific means μ i +1 0 and μ i +1 1 are simulated from this posterior. d. In our specifications, we have worked with only one quarterly variable, GDP growth. We fix the common-factor loading γ 1 for the quarterly GDP growth to be equal to one for identification purposes—this assumption amounts to scaling the common factor. Then, the factor loadings for the monthly variables are interpreted as relative to the unit loading for the GDP growth. In the prior, a factor loading γ j of a monthly indicator j = 1 , ..., M is normally distributed: γ j ∼ N ( a, V ). Then, define ˜ y j,t and ˜ f j,t as follows: ˜ y j,t = y j,t - ψ i j, 1 y j,t - 1 - ... - ψ i j,P y j,t - P ; ˜ f i j,t = f i t - ψ i j, 1 f i t - 1 - ... - ψ i j,P f i t - P . We can use these definitions together with equations ( A.3 ) and ( A.4 ) to derive the following expression: ˜ y j,t = γ j ˜ f i j,t + e j,t , e j,t ∼ N (0 , σ 2 j ) . Using this expression, we can find the posterior for the factor loading to be normal ECB Working Paper Series No 2381 / March 2020 44
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as well: γ j ∼ N a j , ¯ V j ), such that ¯ V j = V - 1 + ( ˜ X j ) 0 ˜ X j - 1 ; ¯ a j = ¯ V V - 1 a + ( ˜ X j ) 0 ˜ Y j , where ˜ X j and ˜ Y j are vectors with elements { ˜ f j,t } and { ˜ y j,t } defined above. Factor loadings { γ i +1 j } are simulated from these posteriors. e. We assume that the individual component of the only quarterly variable in our model is a white noise, which makes it simpler to compute the posterior, due to the monthly missing observations in a variable at the quarterly frequency: for the individual com- ponent of GDP growth, equation ( A.4 ) reduces to u 1 ,t = e 1 ,t , e 1 ,t ∼ N (0 , σ 2 1 ) . Then, we specify the inverse-gamma prior σ 2 1 ∼ IG ( a, b ), which conjugates with the inverse-gamma posterior IG a, ¯ b ), such that ¯ a = a + T 2 , 1 b + ( y i 1 ,t ) 2 / 2 . The variance ( σ i +1 1 ) 2 is simulated from this posterior. f. Define Y j = ( y j, 1 , ..., y j,T ) to be the vector collecting the observations of a monthly variable j . Let X j be the T × P matrix recording the P lags of the variable y j,t . For the AR coefficients of each monthly variable’s individual component, ψ j = ( ψ j, 1 , ..., ψ j,P ) 0 , we assume the same normal prior: ψ j ∼ N ( a, V ). Then, the posterior is also normal, N a j , ¯ V j ), but different for each j , such that ¯ V j = V - 1 a + X 0 j X j ( σ i j ) 2 ! - 1 ; ¯ a j = ¯ V j V - 1 a + X 0 j Y j ( σ i j ) 2 ! We simulate ψ i +1 j from this posterior. Finally, to simulate σ i +1 j , we assume inverse- ECB Working Paper Series No 2381 / March 2020 45
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gamma prior: for each j , σ 2 j ∼ IG ( a, b ). Let Ψ i +1 j be the following P × P matrix: Ψ i +1 j = ψ i +1 j, 1 · · · ψ i +1 j,P - 1 ψ i +1 j,P 1 · · · 0 0 . . . . . . . . . . . . 0 · · · 1 0 Then, we can express the posterior of σ 2 j as inverse-gamma distribution as well, IG a, ¯ b ), such that ¯ a = a + T 2 ; ¯ b = b + ( Y j - X j Ψ i +1 j ) 0 ( Y j - X j Ψ i +1 j ) 2 !
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  • Fall '19
  • Economics, Recession, Late-2000s recession, GWI

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