Technical Appendix: Imperfect information and the business cycle

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1 Technical Appendix: Imperfect information and the usiness cycle Farice Collard Harris Dellas Frank Smets March 29 We would like to thank Marty Eichenaum, Jesper Lindé, Thomas Luik, Frank Schorfheide and Raf Wouters for valuale suggestions. School of Economics, The University of Adelaide, SA 55 Australia. Tel: (+61 ( Fax: (+61 ( URL: Homepage: Department of Economics, University of Bern, CEPR. Address: VWI, Schanzeneckstrasse 1, CH 312 Bern, Switzerland. Tel: ( , Fax: ( , Homepage: Frank Smets: European Central Bank, CEPR and Ghent University, Kaiderstrasse 29 D-6311 Frankfurt am Main, Germany, Tel: ( Fax: (

2 1 Roustness Analysis Tale 1: Diffuse Priors Param. Type Param 1 Param 2 95% HPDI ϑ Uniform. 1. [.25;.975] ξ Uniform. 1. [.25;.975] ϕ Uniform. 1. [.25;.975] r Uniform. 4. [.1;3.9] π Uniform. 4. [.1;3.9] ρ r Uniform. 1. [.25;.975] α π Normal [.52;2.47] α y Normal [.27;.222] ρ a Uniform. 1. [.25;.975] ρ χ Uniform. 1. [.25;.975] ρ π Uniform.5.5 [.25;.975] σ a Invgamma.2 4. [.1;.38] σ χ Invgamma.2 4. [.1;.38] σ r Invgamma.2 4. [.1;.38] σ π Invgamma.2 4. [.1;.38] σ ν Invgamma.2 4. [.1;.38] η y Invgamma.2 4. [.1;.38] η π Invgamma.2 4. [.1;.38] Note: The parameters are distriuted independently from each other. a 95-percent highest proaility density (HPD credile intervals (see?, p.57. The Param 1 and Param 2 report the lower and upper ounds for Uniform distriutions, the mean and the standard deviation for the Normal distriutions. They report the s and ν parameters of the inverse gamma distriution, where f(σ s, ν σ (1+ν exp( νs 2 /2σ 2. 2

3 Tale 2: Moments: Data linearly detrended Data (1 (2 (3 (4 (5 σ y [3.11,9.86] [2.49,5.94] [2.95,1.39] [3.13,12.8] [2.,3.9] σ π [.45,.68] [.57,.85] [.44,.67] [.48,.81] [.41,.61] σ R [.55,1.] [.7,1.1] [.53,.97] [.63,1.33] [.48,.74] ρ(π, y [-.35,-.4] [-.26,.37] [-.32,-.3] [-.1,-.2] [-.15,.15] ρ(r, y [-.52,-.12] [-.65,-.5] [-.5,-.12] [-.23,-.4] [-.17,.27] ρ y ( [.94,1.] [.95,.99] [.94,1.] [.97,1.] [.9,.96] ρ π ( [.56,.82] [.83,.92] [.57,.83] [.71,.9] [.66,.85] ρ R ( [.85,.96] [.91,.96] [.84,.96] [.89,.98] [.85,.94] ρ y ( [.89,.99] [.87,.98] [.89,1.] [.94,1.] [.81,.91] ρ π ( [.39,.74] [.67,.84] [.39,.73] [.61,.87] [.6,.8] ρ R ( [.74,.92] [.8,.91] [.73,.91] [.77,.95] [.71,.87] ρ y ( [.81,.99] [.7,.93] [.82,.99] [.89,1.] [.67,.84] ρ π ( [.25,.64] [.4,.69] [.26,.65] [.46,.8] [.51,.73] ρ R ( [.58,.86] [.55,.8] [.55,.84] [.59,.9] [.5,.73] Note: (1: Baseline NK, (2: Hyrid NK (Backward Indexation, Real Rigidities, (3: Imperfect Info. Temporary vs Permanent Shocks, (4: Imperfect Info., Cogley Sordone, (5: Imperfect Info., Noisy Signals. 3

4 2 Detailed Tales, Tale 3: Posteriors Perfect Info, Forward NK ξ [.33,.64] ϕ [.4,.48] r [.31,.86] π [.53, 1.24] ρ r [.6,.75] α π [ 1.29, 1.9] α y [.4,.23] ρ a [.96, 1.] ρ χ [.85,.97] σ a [.14,.44] σ χ [.11,.22] σ r [.3,.41] σ ν [.1,.19] Average log marginal density:

5 Tale 4: Posteriors Perfect Info, Hyrid NK θ [.9,.97] ξ [.3,.14] ϕ [.7,.51] r [.36,.97] π [.69, 1.33] ρ r [.77,.91] α π [ 1.6, 1.75] α y [.,.12] ρ a [.,.18] ρ χ [.9,.42] σ a [.8,.14] σ χ [.48,.77] σ r [.23,.29] σ ν [.11,.18] Average log marginal density: Tale 5: Posteriors Imperfect Info, Pers. vs Temp. ξ [.34,.64] ϕ [.5,.49] r [.34,.87] π [.57, 1.26] ρ r [.6,.74] α π [ 1.31, 1.9] α y [.5,.24] ρ a [.96, 1.] ρ χ [.85,.97] σ a [.12,.4] σ χ [.12,.23] σ r [.3,.41] σ ν [.11,.22] Average log marginal density:

6 Tale 6: Posteriors Imperfect info, Inflation target shock ξ [.51,.83] ϕ [.2,.46] r [.11,.92] π [.34, 1.3] ρ r [.2,.39] α π [ 2.5, 2.7] α y [.3,.22] ρ a [.97, 1.] ρ χ [.85,.98] ρ π [.86,.98] σ a [.35, 1.38] σ χ [.16,.24] σ r [.37,.63] σ π [.8,.15] Average log marginal density: Tale 7: Posteriors Imperfect info, Measurement errors ξ [.14,.31] ϕ [.6,.5] r [.22, 1.2] π [.57, 1.34] ρ r [.14,.42] α π [ 1.19, 2.1] α y [.11,.28] ρ a [.92,.97] ρ χ [.8,.92] σ a [.9,.16] σ χ [.21,.37] σ r [.9,.16] σ ν [.18,.25] η y [.1,.52] η π [ 2.18, 13.15] Average log marginal density:

7 2.1 Detailed Tales, More diffuse priors Tale 8: Posteriors Perfect Info, Forward NK ξ [.35,.77] ϕ [.1,.93] r [.36,.94] π [.59,1.32] ρ r [.6,.75] α π [1.29,2.2] α y [.2,.28] ρ a [.96,1.] ρ χ [.86,.98] σ a [.14,.48] σ χ [.11,.21] σ r [.3,.42] σ ν [.1,.19] Average log marginal density:

8 Tale 9: Posteriors Perfect Info, Hyrid NK θ [.9,.98] ξ [.3,.17] ϕ [.12, 1.] r [.4,.99] π [.73, 1.34] ρ r [.77,.92] α π [ 1., 1.83] α y [.,.9] ρ a [.,.15] ρ χ [.7,.4] σ a [.8,.14] σ χ [.49,.78] σ r [.23,.3] σ ν [.1,.18] Average log marginal density: Tale 1: Posteriors Imperfect Info, Pers. vs Temp. ξ [.34,.77] ϕ [.6,.98] r [.37,.93] π [.6, 1.31] ρ r [.6,.76] α π [ 1.2, 1.96] α y [.4,.31] ρ a [.97, 1.] ρ χ [.85,.97] σ a [.11,.41] σ χ [.11,.23] σ r [.3,.42] σ ν [.11,.22] Average log marginal density:

9 Tale 11: Posteriors Imperfect info, Inflation target shock ξ [.56,.91] ϕ [.,.74] r [.15, 1.16] π [.39, 1.47] ρ r [.,.37] α π [ 2.47, 3.61] α y [.2,.26] ρ a [.97, 1.] ρ χ [.86,.98] ρ π [.89,.99] σ a [.44, 1.71] σ χ [.18,.25] σ r [.43,.78] σ π [.8,.14] Average log marginal density: Tale 12: Posteriors Imperfect info, Measurement errors ξ [.15,.42] ϕ [.11, 1.] r [.28, 1.11] π [.59, 1.34] ρ r [.14,.44] α π [ 1., 2.45] α y [.11,.35] ρ a [.92,.97] ρ χ [.8,.93] σ a [.9,.16] σ χ [.22,.38] σ r [.9,.17] σ ν [.18,.25] η y [.11,.53] η π [ 2.8, 12.47] Average log marginal density:

10 2.2 Detailed Tales, Alternative Specifications Tale 13: Posteriors Perfect Info, Hyrid NK (Partial indexation to the inflation target θ [.88,.97] ξ [.3,.12] γ [.79,.97] ϕ [.7,.51] r [.42,.92] π [.75,1.27] ρ r [.64,.86] α π [1.1,1.47] α y [.,.3] ρ a [.,.19] ρ χ [.12,.45] ρ π [.,.22] σ a [.9,.16] σ χ [.45,.74] σ r [.17,.29] σ π [.9,.16] Average log marginal density:

11 Tale 14: Posteriors Imperfect info, Measurement errors (R and π are perfectly oservale ξ [.34,.64] ϕ [.5,.5] r [.37,.89] π [.54,1.24] ρ r [.6,.74] α π [1.27,1.88] α y [.5,.24] ρ a [.96,1.] ρ χ [.84,.96] σ a [.12,.39] σ χ [.12,.26] σ r [.3,.41] σ ν [.1,.2] η y [.1,.38] Average log marginal density:

12 3 Detailed Tales, Post 1982 Tale 15: Posteriors Perfect Info, Forward NK (Post 82 period ξ [.57,.91] ϕ [.2,.46] r [.8,1.46] π [.,1.4] ρ r [.1,.61] α π [1.98,2.55] α y [.4,.23] ρ a [.97,1.] ρ χ [.95,1.] σ a [.28,2.22] σ χ [.1,.15] σ r [.25,.48] σ ν [.9,.21] Average log marginal density:

13 Tale 16: Posteriors Perfect Info, Hyrid NK (Post 82 period θ [.76,.99] ξ [.2,.27] ϕ [.6,.51] r [.34,1.2] π [.39,1.14] ρ r [.69,.9] α π [1.8,1.99] α y [.1,.18] ρ a [.,.74] ρ χ [.36,.86] σ a [.7,.14] σ χ [.12,.41] σ r [.16,.24] σ ν [.1,.17] Average log marginal density: Tale 17: Posteriors Imperfect Info, Pers. vs Temp. (Post 82 period ξ [.45,.85] ϕ [.3,.46] r [.9,1.45] π [.1,1.8] ρ r [.24,.65] α π [1.9,2.48] α y [.5,.24] ρ a [.97,1.] ρ χ [.95,1.] σ a [.15,1.3] σ χ [.1,.15] σ r [.24,.43] σ ν [.11,.27] Average log marginal density:

14 Tale 18: Posteriors Imperfect info, Inflation target shock (Post 82 period ξ [.5,.83] ϕ [.2,.46] r [.7,1.44] π [.,1.4] ρ r [.6,.53] α π [1.96,2.54] α y [.4,.22] ρ a [.97,1.] ρ χ [.95,1.] ρ π [.1,.55] σ a [.27,1.9] σ χ [.1,.15] σ r [.11,.32] σ π [.13,.26] Average log marginal density: Tale 19: Posteriors Imperfect info, Measurement errors (Post 82 period ξ [.1,.26] ϕ [.7,.51] r [.6, 1.33] π [.46, 1.4] ρ r [.2,.56] α π [ 1.3, 2.17] α y [.1,.27] ρ a [.84,.94] ρ χ [.87,.96] σ a [.7,.13] σ χ [.16,.29] σ r [.9,.15] σ ν [.13,.2] η y [.11,.52] η π [.87, 8.78] Average log marginal density:

15 Tale 2: Moments (Post 82 period Data (1 (2 (3 (4 (5 σ y [2.2,9.81] [1.82,6.14] [2.34,1.6] [2.3,9.92] [1.36,2.19] σ π [.32,1.1] [.42,.97] [.31,1.5] [.32,1.12] [.29,.46] σ R [.58,2.43] [.65,1.22] [.55,2.25] [.58,2.51] [.46,.86] ρ(π, y [-.13,-.1] [-.21,.72] [-.14,-.1] [-.12,-.1] [.7,.55] ρ(r, y [-.17,-.2] [-.42,.42] [-.18,-.2] [-.17,-.2] [.14,.69] ρ y ( [.97,1.] [.94,1.] [.96,1.] [.97,1.] [.85,.94] ρ π ( [.69,.99] [.79,.96] [.73,.99] [.74,.99] [.61,.84] ρ R ( [.94,1.] [.91,.98] [.94,1.] [.95,1.] [.9,.97] ρ y ( [.94,1.] [.82,.99] [.93,1.] [.94,1.] [.73,.89] ρ π ( [.63,.99] [.58,.91] [.65,.98] [.65,.99] [.53,.78] ρ R ( [.9,1.] [.79,.94] [.89,1.] [.9,1.] [.79,.94] ρ y ( [.88,1.] [.52,.95] [.88,1.] [.88,1.] [.55,.79] ρ π ( [.56,.98] [.25,.81] [.58,.98] [.57,.98] [.41,.69] ρ R ( [.81,1.] [.5,.82] [.8,.99] [.81,1.] [.59,.86] Note: (1: Baseline NK, (2: Hyrid NK (Backward Indexation, Real Rigidities, (3: Imperfect Info. Temporary vs Permanent Shocks, (4: Imperfect Info., Cogley Sordone, (5: Imperfect Info., Noisy Signals. 95% HPDI in rackets. 15

16 Tale 21: HP filtered Moments (Post 82 period Data (1 (2 (3 (4 (5 σ y [.81,1.13] [1.14,2.32] [.83,1.17] [.84,1.17] [.81,1.1] σ π [.2,.27] [.33,.53] [.19,.26] [.2,.27] [.2,.27] σ R [.25,.33] [.39,.65] [.25,.34] [.25,.34] [.26,.36] ρ(π, y [-.3,.15] [.2,.65] [-.3,.16] [.2,.2] [.14,.35] ρ(r, y [-.21,-.3] [-.38,.27] [-.27,-.5] [-.2,-.2] [.5,.47] ρ y ( [.69,.72] [.85,.94] [.64,.72] [.69,.72] [.64,.7] ρ π ( [.11,.32] [.67,.81] [.17,.41] [.23,.44] [.23,.45] ρ R ( [.61,.72] [.81,.91] [.61,.73] [.67,.77] [.69,.81] ρ y ( [.45,.48] [.61,.79] [.4,.48] [.45,.48] [.37,.46] ρ π ( [.3,.14] [.37,.61] [.5,.19] [.6,.2] [.14,.28] ρ R ( [.39,.48] [.58,.75] [.39,.49] [.44,.52] [.44,.57] ρ y ( [.11,.12] [.11,.4] [.9,.12] [.11,.12] [.2,.1] ρ π ( [-.4,.] [-.12,.19] [-.4,.1] [-.6,-.1] [-.1,.4] ρ R ( [.9,.12] [.9,.38] [.8,.13] [.1,.14] [.6,.15] Note: (1: Baseline NK, (2: Hyrid NK (Backward Indexation, Real Rigidities, (3: Imperfect Info. Temporary vs Permanent Shocks, (4: Imperfect Info., Cogley Sordone, (5: Imperfect Info., Noisy Signals. 95% HPDI in rackets. 16

17 3.1 Figures, Post 1982 period 17

18 Figure 1: Impulse Response Functions Perfect info, forward NK (Post 82 period (a Technology Shock ( Preference Shock (c Interest Rate Shock (d Cost Push Shock

19 Figure 2: Impulse Response Functions Perfect info, hyrid NK (Post 82 period (a Technology Shock ( Preference Shock (c Interest Rate Shock (d Cost Push Shock

20 Figure 3: Impulse Response Functions Imperfect info, Pers. vs Temp. (Post 82 period (a Technology Shock ( Preference Shock (c Interest Rate Shock (d Cost Push Shock

21 Figure 4: Impulse Response Functions Imperfect info, Inflation target shock (Post 82 period (a Technology Shock ( Preference Shock (c Interest Rate Shock (d Inflation Target Shock

22 Figure 5: Impulse Response Functions Imperfect info, measurement errors (Post 82 period (a Technology Shock ( Preference Shock (c Interest Rate Shock (d Cost Push Shock

23 4 Solution Method Let the state of the economy e represented y two vectors X t and X f t. The first one includes the predetermined (ackward looking state variales, i.e. X t = ( R t 1, z t, g t, ε R t, whereas the second one consists of the forward looking state variales, i.e. Xf t the following representation M ( X t+1 E t Xf t+1 + M 1 ( X t X f t = (ỹ t, π t. The model admits = M 2 ε t+1 (1 Let us denote the signal process y {S t }. The measurement equation relates the state of the economy to the signal: S t = C ( X t X f t + v t. (2 Finally u and v are assumed to e normally distriuted covariance matrices Σ uu and Σ vv respectively and E(uv =. X t+i t = E(X t+i I t for i and where I t denotes the information set availale to the agents at the eginning of period t. The information set availale to the agents consists of i the structure of the model and ii the history of the oservale signals they are given in each period: I t = {S t j, j, M, M 1, M 2, C, Σ uu, Σ vv } The information structure of the agents is descried fully y the specification of the signals. 4.1 Solving the system Step 1: We first solve for the expected system: M ( X t+1 t X f t+1 t + M 1 ( X t t X f t t = (3 which rewrites as where ( X t+1 t X f t+1 t = W ( X t t W = M 1 M 1 After getting the Jordan form associated to (4 and applying standard methods for eliminating ules, we get From which we get X f t t = GX t t X f t t X t+1 t = (W + W f GX t t = W X t t (5 X f t+1 t = (W f + W ff GX t t = W f X t t (6 (4 23

24 Step 2: We have M ( X t+1 X f t+1 t + M 1 ( X t X f t = M 2 u t+1 Taking expectations, we have M ( X t+1 t X f t+1 t + M 1 ( X t t X f t t = Sutracting, we get M ( X t+1 X t+1 t + M 1 ( X t X t t X f t Xf t t = M 2 u t+1 (7 which rewrites ( X t+1 X t+1 t = W c ( X t X t t X f t Xf t t + M 1 M 2u t+1 (8 where, W c = M 1 M 1. Hence, considering the second lock of the aove matrix equation, we get which gives W c f (X t X t t + W c ff (Xf t Xf t t = X f t = F X t + F 1 X t t with F = W c ff 1 W c f and F 1 = G F. Now considering the first lock, we have from which we get, using (5 X t+1 = X t+1 t + W c (X t X t t + W c f (Xf t Xf t t + M 2 u t+1 X t+1 = M X t + M 1 X t t + M 2 u t+1 with M = W c + W c f F, M 1 = W M and M 2 = M 1 M 2. We also have S t = C Xt + C f X f t + v t from which we get S t = S Xt + S 1 Xt t + v t where S = C + C f F and S 1 = C f F 1 24

25 4.2 Filtering Since our solution involves terms in Xt t, we would like to compute this quantity. However, the only information we can exploit is a signal S t that was descried previously. We therefore use a Kalman filter approach to compute the optimal prediction of X t t. In order to recover the Kalman filter, it is a good idea to think in terms of expectation errors. Therefore, let us define and X t = X t X t t 1 S t = S t S t t 1 Note that since S t depends on Xt t, only the signal relying on S t = S t S 1 Xt t can e used to infer anything on Xt t. Therefore, the policy maker revises its expectations using a linear rule depending on S t e = S t S 1 Xt t. The filtering equation then writes X t t = X t t 1 + K( S e t S e t t 1 = X t t 1 + K(S X t + v t where K is the filter gain matrix, that we would like to compute. The first thing we have to do is to rewrite the system in terms of state space representation. Since S t t 1 = (S + S 1 Xt t 1, we have S t = S (X t X t t + S1 (X t t X t t 1 + v t = S X t + S 1 K(S X t + v t + v t = S X t + ν t where S = (I + S 1 KS and ν t = (I + S 1 Kv t. Now, consider the law of motion of ackward state variales, we get X t+1 = M (Xt Xt t + M 2 u t+1 = M (Xt Xt t 1 X t t + X t t 1 + M 2 u t+1 = M X t M (Xt t + X t t 1 + M 2 u t+1 = M X t M K(S X t + v t + M 2 u t+1 = M X t + ω t+1 where M = M (I KS and ω t+1 = M 2 u t+1 M Kv t. We therefore end up with the following state space representation X t+1 = M X t + ω t+1 (9 S t = S X t + ν t (1 25

26 For which the Kalman filter is given y X t t = X t t 1 + P S (S P S + Σ νν 1 (S X t + ν t But since X t t is an expectation error, it is not correlated with the information set in t 1, such that X t t 1 =. The prediction formula for X t t therefore reduces to X t t = P S (S P S + Σ νν 1 (S X t + ν t (11 where P solves P = M P M + Σ ωω and Σ νν = (I + S 1 KΣ vv (I + S 1 K and Σ ωω = M KΣ vv K M + M 2 Σ uu M 2 Note however that the aove solution is otained for a given K matrix that remains to e computed. We can do that y using the asic equation of the Kalman filter: Solving for Xt t, we get X t t = X t t 1 + K( S e t S e t t 1 = X t t 1 + K(S t S 1 X t t (S t t 1 S 1 X t t 1 = X t t 1 + K(S t S 1 X t t S X t t 1 X t t = (I + KS 1 1 (X t t 1 + K(S t S X t t 1 = (I + KS 1 1 (X t t 1 + KS1 X t t 1 KS1 X t t 1 + K(S t S X t t 1 = (I + KS 1 1 (I + KS 1 X t t 1 + (I + KS1 1 K(S t (S + S 1 X t t 1 = X t t 1 + (I + KS1 1 K S t = X t t 1 + K(I + S1 K 1 St = X t t 1 + K(I + S1 K 1 (S X t + ν t where we made use of the identity (I + KS 1 1 K K(I + S 1 K 1. Hence, identifying to (11, we have K(I + S 1 K 1 = P S (S P S + Σ νν 1 rememering that S = (I + S 1 KS and Σ νν = (I + S 1 KΣ vv (I + S 1 K, we have K(I+S 1 K 1 = P S (I+S 1 K ((I+S 1 KS P S (I+S 1 K +(I+S 1 KΣ vv (I+S 1 K 1 (I+S 1 KS which rewrites as [ K(I + S 1 K 1 = P S (I + S 1 K (I + S 1 K(S P S + Σ vv (I + S 1 K ] 1 K(I + S 1 K 1 = P S (I + S 1 K (I + S 1 K 1 (S P S + Σ vv 1 (I + S 1 K 1 26

27 Hence, we otain K = P S (S P S + Σ vv 1 (12 Now, recall that P = M P M + Σ ωω Rememering that M = M (I + KS and Σ ωω = M KΣ vv K M + M 2 Σ uu M 2, we have P = M (I KS P [ M (I KS ] + M KΣ vv K M + M 2 Σ uu M 2 [ = M (I KS P (I S K + KΣ vv K ] M + M 2 Σ uu M 2 Plugging the definition of K in the latter equation, we otain [ ] P = M P P S (S P S + Σvv 1 S P M + M 2 Σ uu M 2 ( Summary We end up with the system of equations: Xt+1 = M Xt + M 1 Xt t + M 2 u t+1 (14 S t = S X t + S 1 X t t + v t (15 X f t = F Xt + F 1 Xt t (16 Xt t = Xt t 1 + K(S (Xt Xt t 1 + v t (17 Xt+1 t = (M + M 1 Xt t (18 which fully descrie the dynamics of our economy. This may e recast as a standard state space prolem Xt+1 t+1 = Xt+1 t + K(S (Xt+1 Xt+1 t + v t+1 = (M + M 1 Xt t + K(S (M Xt + M 1 Xt t + M 2 u t+1 (M + M 1 Xt t + v t+1 = KS M Xt + ((I KS M + M 1 Xt t + KS M 2 u t+1 + Kv t+1 Then ( X t+1 Xt+1 t+1 where and where = M x ( X t X t t + M e ( ut+1 v t+1 ( M M M x = 1 KS M ((I KS M + M 1 and M e = X f t = M f ( X t X t t M f = ( F F 1 ( M 2 KS M 2 K 27