Ecb working paper series no 2381 march 2020 41 going

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ECB Working Paper Series No 2381 / March 2020 41
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Going with t backwards from T - 1 to 1, compute the following: Forecast and forecast error: ˆ Ω t +1 | t = ˆ FP t | t ( ˆ F ) 0 + ˆ Q ; ˆ z t +1 | t = ˆ Fz t | t + h s i t +1 ( μ 0 + x i t +1 ) + (1 - s i t +1 ) μ 1 , 0 , ..., 0 i 0 ; ν t +1 = ˆ z i +1 t +1 - ˆ z t +1 | t ; Use the information from t + 1 to update the estimates for t : z t | T = z t | t +1 = z t | t + P t | t ( ˆ F ) 0 ( ˆ Ω t +1 | t ) - 1 ν t +1 ; P t | T = P t | t +1 = P t | t - P t | t ( ˆ F ) 0 ( ˆ Ω t +1 | t ) - 1 ˆ FP t | t ; Use the obtained information to randomize z i +1 t ∼ N ( z t | T , P t | T ); Eliminate, as described above, elements of z i +1 t to obtain ˆ z i +1 t . 2. Given Z i +1 , X i , and θ i , operate upon the equation ( A.1 ) to simulate indicators S i +1 following Carter and Kohn ( 1994 ): a. Going forwards, for t = 1 , .., T , compute P t ( s t = 0): The initial unconditional probability of normal state is P 0 ( s 0 = 0) = 1 - q 2 - q - p . For t = 1 , ..., T , first compute P t - 1 ( s t = 0) = P t - 1 ( s t - 1 = 0) × p + ( 1 - P t - 1 ( s t - 1 = 0) ) × (1 - q ); and observe that f t | s t = 0 ∼ N ( μ 0 , σ 2 e ) , f t | s t = 1 ∼ N ( μ 1 + x i t , σ 2 e ) . Then, P t ( s t = 0) = P t - 1 ( s t = 0) φ f t - μ 0 σ e P t - 1 ( s t = 0) φ f t - μ 0 σ e + ( 1 - P t - 1 ( s t = 0) ) φ f t - μ 1 - x i t σ e . ECB Working Paper Series No 2381 / March 2020 42
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b. Going backwards for t = T, ..., 1, compute P t +1 ( s t = 0) = P T ( s t = 0) and simulate the state indicators using these probabilities: For the last period, we have P T ( s T = 0) from step (a). Simulate s i +1 T using this probability. For t = T - 1 , ..., 1, compute the probabilities as follows: s i +1 t +1 = 0 P t +1 ( s t = 0) = P t ( s t = 0) × p P t ( s t = 0) × p + ( 1 - P t ( s t = 0) ) × (1 - q ) ; s i +1 t +1 = 1 P t +1 ( s t = 0) = P t ( s t = 0) × (1 - p ) P t ( s t = 0) × (1 - p ) + ( 1 - P t ( s t = 0) ) × q . Then, since P t +1 ( s t = 0) = P T ( s t = 0), use this probability to simulate s i +1 t 3. Given, S i +1 , θ i , and Z i +1 , simlulate X i +1 using Carter and Kohn ( 1994 ). It is the same routine as in step 1, except that now, the dynamics of the unobserved state are given by equation ( A.2 ), and the “measurement” equation is equation ( A.1 ) for the common factor, in which all the elements except for x t are fixed. A major simplification is that the common factor values are known for all periods t = 1 , ..., T , and both the common factor f t and the latent variable x t are one-dimensional, so there is no need to reduce the dimensionality of the equations to account for missing variables or singularity. 4. Finally, given Y , S i +1 , Z i +1 , and X i +1 , compute θ i +1 using standard pior distributions: a. The prior distribution for σ 2 v , the variance of the shock affecting the unobserved process x t , is inverse-gamma: σ 2 v ∼ IG ( a, b ). Then, the posterior is also inverse- gamma, IG a, ¯ b ), such that ¯ a = a + T 2 , ¯ b = b + ( x i t - s i t x i t - 1 ) 2 2 . We sample ( σ i +1 v ) 2 from this posterior. b. Similarly, the prior distribution for σ 2 e , the variance of the shock affecting the common factor f t , is inverse-gamma: σ 2 e ∼ IG ( a, b ). Then, the posterior is also inverse-gamma, IG a, ¯ b ), such that ¯ a = a + T 2 , ¯ b = 1 b + ∑ ( f i t - s i t ( μ i 1 + x i t - 1 ) - (1 - s i t ) μ i 0 ) 2 / 2 . ECB Working Paper Series No 2381 / March 2020 43
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We sample ( σ i +1 e ) 2 from this posterior.
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