RDP 2010-02: Learning in an Estimated Small Open Economy Model Appendix C: Robustness Checks

Jarkko Jääskelä and Rebecca McKibbin

March 2010

Table C1: Looser Prior and Posterior Distributions for Rational Expectations Model
Parameters Prior Posterior
Distribution Mean Std dev Mean Std dev 5% 95%
Households and firms
β   0.99     0.99      
γ Gamma 1.20 0.20   1.075 0.212 0.971 1.667
η Uniform [0,1)     0.830 0.057 0.730 0.911
φ Gamma 2.00 0.40   1.411 0.366 1.249 2.441
ω Beta 0.20 0.10   0.893 0.175 0.290 0.934
δ Gamma 1.50 0.10   1.593 0.102 1.399 1.737
δx Gamma 1.50 0.10   1.311 0.101 1.306 1.637
θd Beta 0.60 0.10   0.692 0.074 0.651 0.901
θm Beta 0.60 0.10   0.811 0.081 0.647 0.913
ϕa Gamma 0.10 0.05   0.166 0.057 0.064 0.243
Taylor rule
ρr Beta 0.75 0.05   0.659 0.047 0.619 0.773
ϕπ Gamma 1.50 0.10   1.544 0.1035 1.390 1.727
ϕy Gamma 0.20 0.10   0.178 0.082 0.081 0.349
Persistence of shocks
ρa Beta 0.60 0.20   0.721 0.079 0.519 0.780
ρs Beta 0.60 0.20   0.558 0.218 0.120 0.951
ρd Beta 0.60 0.20   0.621 0.092 0.520 0.820
ρm Beta 0.60 0.20   0.169 0.038 0.086 0.207
Std dev of shocks (×10−2)
σa Inv gamma 0.1 2   0.071 0.161 0.067 0.466
σs Inv gamma 0.1 2   0.129 0.073 0.059 0.271
σd Inv gamma 0.1 2   0.124 0.139 0.081 0.464
σm Inv gamma 0.1 2   0.059 0.058 0.060 0.233
σr Inv gamma 0.1 2   0.036 0.005 0.027 0.042
Notes: The posterior statistics are based on 2 million draws using the Markov Chains Monte Carlo (MCMC) method with a 25 per cent burn-in period. For the inverse gamma prior distributions, the mode and the degrees of freedom are reported. Measurement errors (et in Equation (19)) are estimated assuming no prior information and are not shown here. The marginal likelihood is −1,569.
Table C2: Looser Prior and Posterior Distributions for Learning Model
Parameters Prior Posterior
Distribution Mean Std dev Mean Std dev 5% 95%
Households and firms
α   0.18     0.18      
β   0.99     0.99      
γ Gamma 1.20 0.20   1.344 0.188 1.048 1.664
η Uniform [0,1)     0.972 0.012 0.950 0.988
φ Gamma 2.00 0.40   1.210 0.275 0.792 1.694
ω Beta 0.20 0.10   0.205 0.053 0.113 0.285
δ Gamma 1.50 0.10   1.638 0.102 1.475 1.813
δx Gamma 1.50 0.10   1.434 0.097 1.280 1.598
θd Beta 0.60 0.10   0.882 0.032 0.825 0.929
θm Beta 0.60 0.10   0.236 0.057 0.153 0.338
ϕa Gamma 0.10 0.05   0.258 0.064 0.162 0.374
ḡ Uniform [0,1)     0.0002 0.0001 0.0001 0.0003
Taylor rule
ρr Beta 0.75 0.05   0.759 0.045 0.684 0.829
ϕπ Gamma 1.50 0.10   1.494 0.096 1.338 1.660
ϕy Gamma 0.20 0.10   0.189 0.090 0.065 0.350
Persistence of shocks
ρa Beta 0.60 0.20   0.643 0.128 0.430 0.848
ρs Beta 0.60 0.20   0.231 0.125 0.081 0.487
ρd Beta 0.60 0.20   0.648 0.112 0.450 0.821
ρm Beta 0.60 0.20   0.953 0.022 0.912 0.983
Std dev of shocks (×10−2)
σa Inv gamma 0.1 2   0.091 0.030 0.053 0.152
σs Inv gamma 0.1 2   0.071 0.022 0.045 0.114
σd Inv gamma 0.1 2   0.252 0.107 0.113 0.450
σm Inv gamma 0.1 2   0.178 0.113 0.065 0.429
σr Inv gamma 0.1 2   0.033 0.005 0.027 0.041
Notes: The posterior statistics are based on 2 million draws using the Markov Chains Monte Carlo (MCMC) method with a 25 per cent burn-in period. For the inverse gamma prior distributions, the mode and the degrees of freedom are reported. Measurement errors (et in Equation (19)) are estimated assuming no prior information and are not shown here. The marginal likelihood is −1,567.
Table C3: Tighter Prior and Posterior Distributions for MSV Learning Model
Parameters Prior Posterior
Distribution Mean Std dev Mean Std dev 5% 95%
Households and firms
α   0.18     0.18      
β   0.99     0.99      
γ Gamma 1.20 0.20   1.327 0.156 1.067 1.605
η Uniform [0,1)     0.977 0.006 0.966 0.985
φ Gamma 2.00 0.40   0.938 0.163 0.658 1.223
ω Beta 0.20 0.05   0.255 0.040 0.193 0.328
δ Gamma 1.50 0.10   1.491 0.083 1.352 1.634
δx Gamma 1.50 0.10   1.504 0.084 1.369 1.655
θd Beta 0.60 0.05   0.733 0.024 0.670 0.771
θm Beta 0.60 0.05   0.605 0.038 0.532 0.661
ϕa Gamma 0.10 0.05   0.224 0.052 0.145 0.322
ḡ Uniform [0,1)     0.0003 0.0001 0.0002 0.0005
Taylor rule
ρr Beta 0.75 0.01   0.750 0.010 0.734 0.766
ϕπ Gamma 1.50 0.10   1.502 0.083 1.366 1.648
ϕy Gamma 0.20 0.10   0.165 0.071 0.070 0.307
Persistence of shocks
ρa Beta 0.60 0.20   0.740 0.131 0.452 0.888
ρs Beta 0.60 0.20   0.389 0.136 0.192 0.659
ρd Beta 0.60 0.20   0.432 0.074 0.337 0.576
ρm Beta 0.60 0.20   0.764 0.081 0.661 0.935
Std dev of shocks (×10−2)
σa Inv gamma 0.1 2   0.077 0.035 0.045 0.067
σs Inv gamma 0.1 2   0.098 0.034 0.058 0.091
σd Inv gamma 0.1 2   3.394 1.632 1.238 3.156
σm Inv gamma 0.1 2   0.077 0.020 0.051 0.073
σr Inv gamma 0.1 2   0.032 0.004 0.026 0.032
Notes: The posterior statistics are based on 2 million draws using the Markov Chains Monte Carlo (MCMC) method with a 25 per cent burn-in period. For the inverse gamma prior distributions, the mode and the degrees of freedom are reported. The marginal likelihood is −1,830.
Table C4: Looser Prior and Posterior Distributions for MSV Learning Model
Parameters Prior Posterior
Distribution Mean Std dev Mean Std dev 5% 95%
Households and firms
α   0.18     0.18      
β   0.99     0.99      
γ Gamma 1.20 0.20   1.295 0.201 0.978 1.643
η Uniform [0,1)     0.894 0.082 0.723 0.977
φ Gamma 2.00 0.40   1.588 0.337 1.081 2.176
ω Beta 0.20 0.10   0.669 0.091 0.506 0.805
δ Gamma 1.50 0.10   1.588 0.103 1.422 1.763
δx Gamma 1.50 0.10   1.463 0.097 1.306 1.627
θd Beta 0.60 0.10   0.765 0.064 0.655 0.864
θm Beta 0.60 0.10   0.696 0.073 0.570 0.809
ϕa Gamma 0.10 0.05   0.293 0.082 0.166 0.437
ḡ Uniform [0,1)     0.0001 0.0001 0.0001 0.0002
Taylor rule
ρr Beta 0.75 0.05   0.754 0.049 0.670 0.830
ϕπ Gamma 1.50 0.10   1.502 0.100 1.341 1.672
ϕy Gamma 0.20 0.10   0.188 0.094 0.065 0.367
Persistence of shocks
ρa Beta 0.60 0.20   0.697 0.127 0.460 0.872
ρs Beta 0.60 0.20   0.788 0.150 0.503 0.968
ρd Beta 0.60 0.20   0.700 0.119 0.479 0.871
ρm Beta 0.60 0.20   0.256 0.121 0.076 0.465
Std dev of shocks (×10−2)
σa Inv gamma 0.1 2   0.143 0.081 0.067 0.121
σs Inv gamma 0.1 2   0.081 0.029 0.048 0.074
σd Inv gamma 0.1 2   0.441 0.265 0.141 0.380
σm Inv gamma 0.1 2   0.044 0.008 0.033 0.043
σr Inv gamma 0.1 2   0.036 0.005 0.027 0.033
Notes: The posterior statistics are based on 2 million draws using the Markov Chains Monte Carlo (MCMC) method with a 25 per cent burn-in period. For the inverse gamma prior distributions, the mode and the degrees of freedom are reported. The marginal likelihood is −1,813.
Figure c1: Impulse Responses to Monetary Shock
Figure C2: Impulse Responses to Productivity Shock