Finance and Economics Discussion Series: 2007-56 Screen Reader version
♣Welfare-Maximizing Monetary Policy
Under Parameter Uncertainty
(Figure Data)
Rochelle M. Edge, Thomas Laubach, and John C. Williams
Figure 1 - Output (y)
| Quarter | VAR impulse response | VAR response plus 1 std dev | VAR response minus 1 std dev | Model impulse response |
| 0 | 0.5769 | 0.8256 | 0.3283 | 0.2710 |
| 1 | 0.5333 | 0.8656 | 0.2010 | 0.4987 |
| 2 | 0.5664 | 0.9300 | 0.2028 | 0.6841 |
| 3 | 0.7930 | 1.1172 | 0.4688 | 0.8306 |
| 4 | 0.9153 | 1.2558 | 0.5748 | 0.9427 |
| 5 | 0.9757 | 1.3289 | 0.6225 | 1.0257 |
| 6 | 0.9773 | 1.3346 | 0.6200 | 1.0845 |
| 7 | 1.0291 | 1.3914 | 0.6668 | 1.1238 |
| 8 | 1.0747 | 1.4508 | 0.6985 | 1.1477 |
Figure 1 - Hours (l)
| Quarter | VAR impulse response | VAR response plus 1 std dev | VAR response minus 1 std dev | Model impulse response |
| 0 | -0.2360 | 0.0127 | -0.4846 | -0.7290 |
| 1 | -0.2585 | 0.0962 | -0.6132 | -0.5013 |
| 2 | -0.2325 | 0.1836 | -0.6487 | -0.3159 |
| 3 | -0.0337 | 0.3905 | -0.4580 | -0.1694 |
| 4 | 0.0086 | 0.4409 | -0.4237 | -0.0573 |
| 5 | 0.0642 | 0.5050 | -0.3766 | 0.0257 |
| 6 | 0.1017 | 0.5396 | -0.3363 | 0.0845 |
| 7 | 0.1293 | 0.5559 | -0.2974 | 0.1238 |
| 8 | 0.1311 | 0.5450 | -0.2827 | 0.1477 |
Figure 1 - Real wage (w)
| Quarter | VAR impulse response | VAR response plus 1 std dev | VAR response minus 1 std dev | Model impulse response |
| 0 | 0.5958 | 1.0146 | 0.1770 | 0.1825 |
| 1 | 0.7611 | 1.2413 | 0.2808 | 0.3339 |
| 2 | 0.9653 | 1.5155 | 0.4151 | 0.4593 |
| 3 | 1.0267 | 1.5837 | 0.4697 | 0.5629 |
| 4 | 1.0204 | 1.5778 | 0.4630 | 0.6482 |
| 5 | 1.0107 | 1.5676 | 0.4537 | 0.7183 |
| 6 | 1.0301 | 1.5966 | 0.4636 | 0.7755 |
| 7 | 1.0411 | 1.6169 | 0.4654 | 0.8222 |
| 8 | 1.0470 | 1.6322 | 0.4618 | 0.8601 |
Figure 1 - Price inflation
| Quarter | VAR impulse response | VAR response plus 1 std dev | VAR response minus 1 std dev | Model impulse response |
| 0 | -0.8296 | -0.4093 | -1.2499 | -0.6526 |
| 1 | -0.6475 | -0.3068 | -0.9882 | -0.5215 |
| 2 | -0.3221 | -0.0215 | -0.6228 | -0.4146 |
| 3 | -0.1514 | 0.1692 | -0.4720 | -0.3278 |
| 4 | -0.3581 | -0.0124 | -0.7037 | -0.2576 |
| 5 | -0.4033 | -0.0735 | -0.7330 | -0.2010 |
| 6 | -0.2636 | 0.0232 | -0.5504 | -0.1556 |
| 7 | -0.0892 | 0.1966 | -0.3749 | -0.1194 |
| 8 | -0.1323 | 0.1573 | -0.4220 | -0.0908 |
Figure 1 - Interest rate
| Quarter | VAR impulse response | VAR response plus 1 std dev | VAR response minus 1 std dev | Model impulse response |
| 0 | -0.1925 | 0.0227 | -0.4078 | -0.1254 |
| 1 | -0.2314 | 0.1563 | -0.6191 | -0.2016 |
| 2 | -0.3305 | 0.1108 | -0.7719 | -0.2427 |
| 3 | -0.3310 | 0.1557 | -0.8178 | -0.2592 |
| 4 | -0.2707 | 0.1786 | -0.7201 | -0.2591 |
| 5 | -0.1743 | 0.2217 | -0.5703 | -0.2481 |
| 6 | -0.1892 | 0.2051 | -0.5834 | -0.2306 |
| 7 | -0.2579 | 0.1303 | -0.6460 | -0.2094 |
| 8 | -0.2341 | 0.1113 | -0.5796 | -0.1868 |
Figure 2 - Output(y)
| Quarter | VAR impulse response | VAR response plus 1 std dev | VAR response minus 1 std dev | Model impulse response |
| 0 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| 1 | -0.1199 | -0.0282 | -0.2116 | -0.2084 |
| 2 | -0.5865 | -0.4559 | -0.7171 | -0.3266 |
| 3 | -0.7305 | -0.5667 | -0.8944 | -0.3833 |
| 4 | -0.7518 | -0.5738 | -0.9298 | -0.3991 |
| 5 | -0.7883 | -0.5984 | -0.9782 | -0.3888 |
| 6 | -0.8012 | -0.5961 | -1.0063 | -0.3628 |
| 7 | -0.7694 | -0.5533 | -0.9855 | -0.3284 |
| 8 | -0.7144 | -0.4914 | -0.9373 | -0.2904 |
Figure 2 - Hours(l)
| Quarter | VAR impulse response | VAR response plus 1 std dev | VAR response minus 1 std dev | Model impulse response |
| 0 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| 1 | 0.0034 | 0.0633 | -0.0564 | -0.2084 |
| 2 | -0.2568 | -0.1565 | -0.3572 | -0.3266 |
| 3 | -0.4476 | -0.3140 | -0.5812 | -0.3833 |
| 4 | -0.5301 | -0.3731 | -0.6870 | -0.3991 |
| 5 | -0.6051 | -0.4320 | -0.7782 | -0.3888 |
| 6 | -0.6349 | -0.4462 | -0.8237 | -0.3628 |
| 7 | -0.6249 | -0.4243 | -0.8255 | -0.3284 |
| 8 | -0.5932 | -0.3844 | -0.8021 | -0.2904 |
Figure 2 - Real wage (w)
| Quarter | VAR impulse response | VAR response plus 1 std dev | VAR response minus 1 std dev | Model impulse response |
| 0 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| 1 | -0.1324 | -0.0745 | -0.1903 | -0.0105 |
| 2 | -0.1490 | -0.0670 | -0.2309 | -0.0190 |
| 3 | -0.1788 | -0.0747 | -0.2828 | -0.0254 |
| 4 | -0.2008 | -0.0888 | -0.3128 | -0.0298 |
| 5 | -0.2136 | -0.0979 | -0.3293 | -0.0324 |
| 6 | -0.2105 | -0.0887 | -0.3323 | -0.0334 |
| 7 | -0.2076 | -0.0801 | -0.3351 | -0.0333 |
| 8 | -0.2043 | -0.0739 | -0.3348 | -0.0323 |
Figure 2 - Price inflation
| Quarter | VAR impulse response | VAR response plus 1 std dev | VAR response minus 1 std dev | Model impulse response |
| 0 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| 1 | 0.1563 | 0.2616 | 0.0510 | -0.0719 |
| 2 | 0.1050 | 0.2256 | -0.0157 | -0.0707 |
| 3 | 0.0169 | 0.1448 | -0.1111 | -0.0681 |
| 4 | -0.1065 | 0.0036 | -0.2166 | -0.0644 |
| 5 | -0.1194 | -0.0043 | -0.2346 | -0.0599 |
| 6 | -0.2290 | -0.1019 | -0.3562 | -0.0550 |
| 7 | -0.2717 | -0.1397 | -0.4037 | -0.0498 |
| 8 | -0.3001 | -0.1755 | -0.4248 | -0.0447 |
Figure 2 - Interest rate
| Quarter | VAR impulse response | VAR response plus 1 std dev | VAR response minus 1 std dev | Model impulse response |
| 0 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
| 1 | 0.9589 | 1.0494 | 0.8685 | 0.7949 |
| 2 | 0.5013 | 0.6340 | 0.3686 | 0.6292 |
| 3 | 0.4286 | 0.5835 | 0.2737 | 0.4958 |
| 4 | 0.3840 | 0.5492 | 0.2187 | 0.3886 |
| 5 | 0.2437 | 0.4079 | 0.0795 | 0.3027 |
| 6 | 0.1543 | 0.3232 | -0.0147 | 0.2343 |
| 7 | 0.0891 | 0.2641 | -0.0858 | 0.1799 |
| 8 | 0.0183 | 0.1950 | -0.1584 | 0.1369 |
Figure 3 - Natural Rate of Hours
| Quarter | Point estimates | Lower 90% CI | Lower 70% CI | Median | Upper 70% CI | Upper 90% CI |
| 1 | -0.2995 | -0.5193 | -0.5064 | -0.4525 | -0.3382 | -0.2411 |
| 2 | -0.0897 | -0.2697 | -0.2564 | -0.2047 | -0.1144 | -0.0582 |
| 3 | -0.0269 | -0.1401 | -0.1298 | -0.0926 | -0.0387 | -0.014 |
| 4 | -0.008 | -0.0727 | -0.0657 | -0.0419 | -0.0131 | -0.0034 |
| 5 | -0.0024 | -0.0378 | -0.0333 | -0.019 | -0.0044 | -0.0008 |
| 6 | -0.0007 | -0.0196 | -0.0169 | -0.0086 | -0.0015 | -0.0002 |
| 7 | -0.0002 | -0.0102 | -0.0085 | -0.0039 | -0.0005 | 0 |
| 8 | -0.0001 | -0.0053 | -0.0043 | -0.0018 | -0.0002 | 0 |
Figure 3 - Natural Rate of Interest
| Quarter | Point estimates | Lower 90% CI | Lower 70% CI | Median | Upper 70% CI | Upper 90% CI |
| 1 | -6.4823 | -7.1148 | -6.072 | -4.1805 | -2.3728 | -1.055 |
| 2 | -1.9416 | -2.1361 | -2.0584 | -1.8218 | -1.151 | -0.4728 |
| 3 | -0.5816 | -0.8991 | -0.8632 | -0.7677 | -0.4681 | -0.1472 |
| 4 | -0.1742 | -0.4187 | -0.3969 | -0.3378 | -0.1664 | -0.0332 |
| 5 | -0.0522 | -0.2034 | -0.189 | -0.1513 | -0.0564 | -0.0073 |
| 6 | -0.0156 | -0.1007 | -0.092 | -0.0677 | -0.0189 | -0.0015 |
| 7 | -0.0047 | -0.0504 | -0.0453 | -0.0305 | -0.0063 | -0.0003 |
| 8 | -0.0014 | -0.0255 | -0.0224 | -0.0136 | -0.002 | -0.0001 |
Figure 4 - Certainty Equivalent Policy
| Quarter | Nominal interest rate (annual) | Output Gap | Price Inflation (annual) | Wage Inflation (annual) |
| 1 | -4.6321 | -0.0129 | -0.6911 | 0.0606 |
| 2 | -2.3062 | -0.0181 | -0.5612 | 0.049 |
| 3 | -1.2023 | -0.0199 | -0.4557 | 0.0396 |
| 4 | -0.6698 | -0.0199 | -0.37 | 0.0321 |
| 5 | -0.4043 | -0.0189 | -0.3004 | 0.0262 |
| 6 | -0.2649 | -0.0172 | -0.2439 | 0.0214 |
| 7 | -0.1863 | -0.0152 | -0.198 | 0.0175 |
| 8 | -0.1383 | -0.0131 | -0.1606 | 0.0144 |
| 9 | -0.1064 | -0.0111 | -0.1303 | 0.0119 |
| 10 | -0.0837 | -0.0091 | -0.1057 | 0.0099 |
| 11 | -0.0668 | -0.0073 | -0.0857 | 0.0083 |
| 12 | -0.0537 | -0.0058 | -0.0694 | 0.0069 |
Figure 4 - Optimal Bayesian Policy.
| Quarter | Nominal interest rate (annual) | Output Gap | Price Inflation (annual) | Wage Inflation (annual) |
| 1 | -5.8387 | 0.0272 | -0.692 | 0.0628 |
| 2 | -2.4988 | 0.0088 | -0.5622 | 0.0478 |
| 3 | -1.1585 | -0.0088 | -0.4569 | 0.0375 |
| 4 | -0.6031 | -0.0183 | -0.3713 | 0.03 |
| 5 | -0.359 | -0.0214 | -0.3017 | 0.0242 |
| 6 | -0.241 | -0.021 | -0.2452 | 0.0196 |
| 7 | -0.1762 | -0.0189 | -0.1993 | 0.0159 |
| 8 | -0.1356 | -0.0163 | -0.162 | 0.0129 |
| 9 | -0.1073 | -0.0137 | -0.1316 | 0.0105 |
| 10 | -0.0861 | -0.0114 | -0.107 | 0.0085 |
| 11 | -0.0695 | -0.0093 | -0.0869 | 0.0069 |
| 12 | -0.0563 | -0.0076 | -0.0707 | 0.0056 |