Gauging Taylor Rules - Do Differences Make a Difference Part 2

Table 1: Data for inflation and output gap: summary (** note on Frequency Vint. and Per.) Output Inflation Variable Source Frequency Vint. Per. Vintages Start End CPI realized GNK SPF PHIL ALFRED PHIL Q 8/y Q M Q Q 1972q3 1979q4 1981q3 2010q4 2005q4 2010q4 CPI core realized GNK SPF ALFRED ALFRED PHIL M 8/y Q M Q Q 1996q1 1986q1 2007q1 2010q4 2005q4 2010q4 PCE realized GNK SPF PHIL ALFRED PHIL Q 8/y Q Q Q Q 1965q4 2000q1 2007q1 2010q4 2005q4 2010q4 PCE core realized GNK SPF PHIL ALFRED PHIL Q 8/y Q Q Q Q 1996q1 2000q1 2007q1 2010q4 2005q4 2010q4 GDP def realized GNK SPF PHIL ALFRED PHIL Q 8/y Q Q Q Q 1965q4 1986q1 1968q4 2010q4 2005q4 2010q4 Output gap GNK PHIL 8/y Q 1987q3 2005q4 Potential CBO ALFRED 1/y 2/y Q Q 1991 1999:1 1998 2009:2 GDP realized GNK SPF PHIL ALFRED PHIL Q 8/y Q Q Q Q 1965q4 1978q1 1968q4 2010q4 2005q4 2010q4 Data for inflation and the output gap come from a variety of sources, depending on whether the data is realized (known as of today) or real-time/ vintage data. Data frequency is mostly quarterly. Not all vintage series span the entire sample of interest, going from 1990 to 2010. ALFRED stands for Archive of Federal Reserve Economic Data, from the Federal Reserve Bank of St. Louis. And PHIL stands for the Real-Time Data Center of the Federal Reserve Bank of Philadelphia. GNK stands for the Federal Reserve Board’s Greenbook. And SPF stands for the Survey of Professional Forecasters. i Table 2: Data for inflation and output gap: availability Output Inflation Variable Vintage 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 CPI Realized GNK SPF CPI core Realized GNK SPF PCE Realized GNK SPF PCE core Realized GNK SPF GDP def Realized GNK SPF Output gap GNK Potential CBO GDP Realized GNK SPF Not all vintage data spans the entire sample of interest, going from 1990 to 2010. See prior table for an explanation of the various acronyms. ii Table 3: RMSE relative to baseline PANEL 1: Contemporaneous, real time data rules Sample Output gap Inflation CPI CPI core PCE PCE core GDP def GNK 1990-2005 CBO HPSPF 0.41 1.17 1.11 base 1.03 0.91 0.99 1.52 1.41 0.99 1.52 1.42 base 1.46 0.90 1.19 1.96 1.55 1.34 1.17 1.52 1.54 1.47 1.87 HPGNK GNK 1990-2010 CBO HPSPF HPGNK PANEL 2: Forward looking, real time data rules Sample Output gap SPF inflation CPI CPI core PCE PCE core GDP def GNK inflation CPI CPI core PCE PCE core GDP def GNK 1990-2005 CBO HPSPF 1.08 0.99 1.31 1.33 1.50 1.91 1.27 1.13 1.41 1.43 1.46 1.83 1.37 1.07 1.25 0.95 1.51 1.27 1.52 1.28 1.42 1.26 1.54 1.56 HPGNK GNK 1990-2010 CBO HPSPF HPGNK PANEL 3: Nowcast, real time data rules Sample Output gap SPF inflation CPI CPI core PCE PCE core GDP def GNK inflation CPI CPI core PCE PCE core GDP def GNK 1990-2005 CBO HPSPF 0.72 0.54 1.11 1.14 0.97 1.48 1.40 1.21 1.56 1.58 1.51 1.91 0.83 1.15 0.72 1.00 1.19 1.48 1.21 1.52 1.36 1.17 1.52 1.54 HPGNK GNK 1990-2010 CBO HPSPF HPGNK PANEL 4: Contemporaneous, realized data rules Sample Output gap Inflation CPI CPI core PCE PCE core GDP def GNK 1990-2005 CBO HPSPF 0.64 1.14 1.18 1.43 1.36 0.99 1.48 1.42 1.69 1.53 HPGNK GNK 1990-2010 CBO HPSPF 0.64 1.43 1.07 1.57 1.56 HPGNK 1.22 1.85 1.53 1.96 1.80 Root mean squared errors relative to baseline. No results are reported for those formulations with insufficient observations to warrant a meaningful comparison to others. Note, GNK = Federal Reserve Board’s Greenbook, CBO = Congressional Budget Office, SPF = Survey of Professional Forecasters, HP = Hodrick-Prescott filter, based on real-time series plus real-time forecasts coming from either SPF or GNK, CPI = Consumer Price Inflation, PCE = Personal Consumer Expenditures, GDP defl = GDP iii deflator, ex-oil = core inflation measure, excluding oil products. iv 1.51 1.65 1.00 0.93 0.92 1.39 1.78 2.06 1.54 1990-2002 CBO HPSPF 1.44 1.61 1.03 GNK 1.40 1.78 2.06 1.54 HPGNK 1.26 1.38 0.85 0.85 0.94 0.88 1.22 1.32 1.44 1.37 1.58 1.67 1.33 1.70 1990-2002 CBO HPSPF 1.37 GNK 1.32 1.50 0.84 0.83 0.82 0.83 1.26 1.43 1.34 1.33 1.60 1.91 1.34 1.65 1990-2002 CBO HPSPF 1.29 GNK 1.35 1.61 1.94 1.36 1.66 HPGNK GNK 1.66 1.79 1.26 1.30 1.29 1.83 2.04 1.47 1.58 1.43 1990-2002 CBO HPSPF HPGNK GNK GNK GNK GNK 2.64 2.02 2.02 1.51 1.88 1.91 2.36 1.93 2.74 2.55 1.93 2.20 1.75 2.55 2.56 1.91 2.21 1.75 2.58 2002-2007 CBO HPSPF 1.47 2.25 2002-2007 CBO HPSPF 1.63 2.02 2002-2007 CBO HPSPF 2.16 1.78 1.61 1.35 1.42 2002-2007 CBO HPSPF HPGNK HPGNK HPGNK HPGNK Root mean squared errors relative to the Federal Funds rate. See Table 3 for details on acronyms. Inflation CPI CPI core PCE PCE core GDP def Sample Output gap PANEL 4: Contemporaneous, realized data rules GNK inflation CPI CPI core PCE PCE core GDP def SPF inflation CPI CPI core PCE PCE core GDP def Sample Output gap 1.37 1.58 1.67 1.34 1.71 HPGNK PANEL 3: Nowcast, real time data rules GNK inflation CPI CPI core PCE PCE core GDP def SPF inflation CPI CPI core PCE PCE core GDP def Sample Output gap PANEL 2: Forward looking, real time data rules Inflation CPI CPI core PCE PCE core GDP def Sample Output gap PANEL 1: Contemporaneous, real time data rules GNK GNK GNK GNK Table 4: RMSE relative to Federal Funds rate 1.84 2.58 2.18 2.18 1.78 2.12 1.79 2.44 2.37 1.72 1.94 1.48 1.78 2.65 2.16 2.28 1.96 2.07 1990-2010 CBO HPSPF 1.10 2.02 1990-2010 CBO HPSPF 1.17 1.57 1990-2010 CBO HPSPF 1.21 2.16 1.57 1.60 1990-2010 CBO HPSPF HPGNK HPGNK HPGNK HPGNK v 2.18 (0.035) - - -2.18 (0.035) 1990-2002 Base (1) Rule (2) - -0.77 (0.445) -1.29 (0.203) -0.02 (0.982) 1990-2002 Rule (2) 0.02 (0.982) - Rule (1) 0.00 - - 0.00 1990-2002 Rule (1) Rule (2) 0.67 (0.510) 1.23 (0.233) - -0.82 (0.418) - - 0.82 (0.418) 2002-2007 Rule (1) Rule (2) - 0.77 (0.445) 1.29 (0.203) Rule (1) - 3.55 (0.002) - 0.93 (0.363) -0.67 (0.510) - 0.69 (0.495) -1.52 (0.133) -0.42 (0.676) - Rule (1) -0.78 (0.437) - 0.42 (0.676) 1990-2010 Rule (2) - 0.78 (0.437) 1.52 (0.133) Rule (3) Giacomini and White (2006) test results, value of test statistic and p-value. Tests compare the statistical significance of the difference between the rule indicated in the row with that indicated in the column, in the ability to forecast the Federal Funds rate one quarter ahead. When the test statistic is positive, the rule indicated in the column is preferred to that in the row. The p-value indicates the significance of the difference in the two rules. -0.69 (0.495) - 1990-2010 Rule (1) Rule (2) -0.93 (0.363) - -1.23 (0.233) - 3.14 (0.002) Rule (3) -3.14 (0.002) - 1990-2010 Base (1) Rule (2) 2002-2007 Rule (2) * Nowcast/GDPdefl/CBO for 1990-2002 and 1990-2010; Contemp/PCEcore/CBO for 2002-2007 (1) Best fitting rule (BFR) * Test statistic (p-value) (2) BFR, optimized assumptions Test statistic (p-value) Sample Rule formulation -3.55 (0.002) - 2002-2007 Base (1) Rule (2) Rule (3) PANEL 3: Best fitting rules and optimized assumptions (1) Contemp/GDP defl/CBO Test statistic (p-value) (2) Fwd/GDP defl/CBO Test statistic (p-value) (3) Nowcast/GDP defl/CBO Test statistic (p-value) Sample Rule formulation PANEL 2: Best fitting rules across different specifications (1) baseline rule Test statistic (p-value) (2) Contemp/GDP defl/CBO Test statistic (p-value) Sample Rule formulation PANEL 1: Contemporary rule specification: baseline versus best fitting rule Table 5: Giacomini and White (2006) test results Table 6: RMSE for different assumptions Panel 1: Best rule sample 1990-2002: Nowcast, GDP deflator, CBO output gap Sample Real interest rate γ1 γ2 0.2 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.5 0.8 0.5 1.0 1.5 2.0 1990-2002 2.5 3.0 3.5 4.0 4.5 5.0 2.254 2.139 2.311 2.269 2.107 2.237 2.302 2.096 2.181 1.838 1.69 1.897 1.863 1.657 1.814 1.911 1.651 1.752 1.476 1.279 1.537 1.515 1.246 1.442 1.581 1.248 1.373 1.216 0.957 1.274 1.273 0.927 1.169 1.361 0.943 1.094 1.13 0.834 1.177 1.203 0.816 1.075 1.305 0.849 1.005 1.541 1.325 1.552 1.611 1.333 1.493 1.704 1.373 1.462 1.917 1.743 1.916 1.98 1.756 1.875 2.063 1.794 1.858 2.339 2.195 2.331 2.397 2.211 2.303 2.471 2.247 2.294 2.788 2.664 2.774 2.841 2.683 2.755 2.908 2.717 2.753 1.255 0.988 1.283 1.331 0.986 1.201 1.433 1.026 1.151 Panel 2: Best rule sample 2002-2007: Contemporaneous, PCE core, CBO output gap Sample Real interest rate γ1 γ2 0.2 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.5 0.8 0.5 1.0 1.5 2.0 2002-2007 2.5 3.0 3.5 4.0 4.5 5.0 1.693 1.948 2.242 1.694 1.966 2.273 1.706 1.994 2.313 1.436 1.62 1.868 1.405 1.614 1.882 1.389 1.622 1.907 1.326 1.398 1.567 1.258 1.359 1.555 1.203 1.335 1.557 1.397 1.335 1.385 1.3 1.261 1.34 1.211 1.199 1.309 1.627 1.454 1.373 1.515 1.353 1.293 1.409 1.261 1.225 2.353 2.067 1.82 2.238 1.953 1.71 2.127 1.843 1.605 2.781 2.47 2.186 2.668 2.357 2.074 2.558 2.247 1.966 3.229 2.904 2.597 3.118 2.792 2.485 3.01 2.684 2.378 3.691 3.356 3.034 3.582 3.246 2.925 3.476 3.14 2.818 1.96 1.716 1.532 1.845 1.604 1.431 1.733 1.498 1.338 Panel 3: Best rule full sample: Nowcast, GDP deflator, CBO output gap Sample Real interest rate γ1 γ2 0.2 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.5 0.8 0.5 1.0 1.5 2.0 1990-2010 2.5 3.0 3.5 4.0 4.5 5.0 1.89 1.927 2.339 1.885 1.909 2.313 1.901 1.912 2.305 1.59 1.529 1.941 1.588 1.511 1.913 1.613 1.52 1.907 1.409 1.21 1.599 1.412 1.193 1.571 1.445 1.211 1.569 1.393 1.043 1.361 1.402 1.031 1.333 1.44 1.059 1.336 1.548 1.101 1.282 1.561 1.096 1.259 1.6 1.13 1.268 2.193 1.721 1.652 2.209 1.727 1.643 2.244 1.757 1.66 2.601 2.141 2.005 2.617 2.15 2.002 2.65 2.178 2.02 3.037 2.59 2.411 3.053 2.6 2.412 3.083 2.626 2.429 3.49 3.055 2.847 3.506 3.066 2.85 3.535 3.091 2.868 1.831 1.355 1.392 1.846 1.357 1.376 1.883 1.39 1.39 Root mean squared errors corresponding to different assumptions on the real interest rate and sensitivity parameters of the best fitting rules in each sub-sample (nowcast, GDP deflator, CBO output gap for the first and full sub-samples and contemporaneous, PCE core, CBO output gap for the middle sub-sample). Lighter cells indicate lower RMSEs. vi Interest rate [in %] Figure 1: Effect of changing inflation measures 10 10 8 8 6 6 4 4 2 2 0 0 -2 -2 90 91 92 93 94 95 96 97 98 99 00 01 Target FFR Taylor rule based on CPI core Taylor rule based on PCE core 02 03 04 05 06 07 08 09 10 Baseline (headline CPI) Taylor rule based on headline PCE Taylor rule based on GDP deflator The effects of changing inflation measures on Taylor interest rates are large. Interest rate [in %] Figure 2: Effect of changing output gap measure 10 10 8 8 6 6 4 4 2 2 0 0 -2 -2 90 91 92 93 94 95 96 97 98 99 00 01 Target FFR Taylor rule based on HP-filtered SPF GDP-forecasts 02 03 04 05 06 07 08 09 10 Baseline (using CBO potential estimation) Taylor rule based on Greenbook output gap The effects of changing output gap measures are more noticeable in the 1990s than in the 2000s. vii Inflation rate (yoy) [in %] Figure 3: Inflation forecasts 6 6 5 5 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -2 90 91 92 93 94 95 96 97 98 99 00 01 Inflation (CPI, as currently measured) 02 03 04 05 SPF forecasts 06 07 08 09 10 11 Greenbook forecasts Four quarters ahead inflation forecasts from both the SPF and Greenbook are good when inflation is steady. But forecasts are rather poor when inflation volatility increases, such as in 1993-94, 1997-98, 2000-02, 2004-06 and again in 2008. It is in these same periods, naturally, that Taylor interest rates from forward-looking and nowcast rules most diverge from the baseline contemporaneous specification. At least, SPF and Greenbook forecasts are very similar during the sample in which they overlap. Interest rate [in %] Figure 4: Effect of rule specification 10 10 8 8 6 6 4 4 2 2 0 0 -2 -2 90 91 92 93 94 95 96 97 98 99 00 Target FFR Forward-looking Taylor rule (SPF forecasts) 01 02 03 04 05 06 07 08 09 10 Baseline (contemporaneous rule) Nowcast Taylor rule (SPF nowcasts) Forward-looking and nowcast rules most diverge from the contemporaneous baseline when inflation forecasts diverge from actual inflation (shown in the prior figure). viii Interest rate [in %] Figure 5: Effect of inflation forecast source 10 10 8 8 6 6 4 4 2 2 0 0 -2 -2 90 91 92 93 94 95 96 97 98 99 00 Target FFR Forward-looking Taylor rule (SPF forecasts) 01 02 03 04 05 06 07 08 09 10 Baseline (contemporaneous rule) Forward-looking Taylor rule (Greenbook forecasts) The effects of forecast source are neither very large nor systematic across the sample. Interest rate [in %] Figure 6: Effect of information set 10 10 8 8 6 6 4 4 2 2 0 0 -2 -2 90 91 92 93 94 95 96 97 98 99 00 Target FFR Realized data 01 02 03 04 05 06 07 08 09 10 Baseline (real-time data) Using either real-time or realized data makes little difference throughout most of the sample. ix Output gap [in %] Figure 7: Revisions to the output gap 6 6 4 4 2 2 0 0 -2 -2 -4 -4 -6 -6 -8 -8 90 91 92 93 94 95 96 97 98 99 00 01 Realised CBO output gap, as measured in 2010Q4 Range of CBO output gap (revisions) through time 02 03 04 05 06 07 08 09 10 11 Real-time CBO output gap, as measured in t-1 Real time CBO output gaps are shown with the dashed line. Realized gaps, instead, appear with the continuous line. The range of revisions having taken place in between is shown with the grey area. The size of the area comes from the fact that revisions are themselves revised, sometimes up, sometimes down. Until the late 1990s, revision of the output gap were generally downward. Then, after the turn of the millennium, output gaps were mostly revised up. In the most recent period, revisions are much smaller, but will grow as time passes. The ultimate revision, between the real time and realized datasets is the difference between the continuous and dotted lines. These were greatest from 1991-92, 1995-99 and 2004-07. Interest rate [in %] Figure 8: Effect of changing natural interest rate 10 10 8 8 6 6 4 4 2 2 0 0 -2 -2 90 91 92 93 94 95 96 97 98 99 00 01 02 03 Target FFR Interest rates from 0.5 to 5%, in steps of 0.5pp 04 05 06 07 08 09 10 Baseline The effects of changing assumptions of the natural interest rate or the inflation target is akin to a level shift in Taylor Rule interest rates. The above captures, equivalently, a change in the natural interest rate between 0.5 and 5%, or a target inflation rate between 6 and 0% and/or a mix thereof (which in many cases would be more realistic). The equivalence between these two sets of changes is captured in the text when re-writing the the Taylor rule after grouping constant terms. x Interest rate [in %] Figure 9: Effect of changing inflation sensitivity 10 10 8 8 6 6 4 4 2 2 0 0 -2 -2 90 91 92 93 94 95 96 97 98 99 00 01 Target FFR Inflation sensitivity equals 0.2 02 03 04 05 06 07 08 09 10 Baseline Inflation sensitivity equals 0.8 Changing policy makers’ sensitivity to deviations of inflation from target has a relatively small effect on recommended interest rates. Interest rate [in %] Figure 10: Effect of changing output gap sensitivity 10 10 8 8 6 6 4 4 2 2 0 0 -2 -2 90 91 92 93 94 95 96 97 98 99 00 Target FFR Output gap sensitivity equals 0.2 01 02 03 04 05 06 07 08 09 10 Baseline Output gap sensitivity equals 0.8 Changing policy makers’ sensitivity to deviations of the output gap has a larger effect, especially in periods when such deviations are large. xi Interest rate [in %] Figure 11: Best-fitting rule for the 1990-2002 and full sample 10 10 8 8 6 6 4 4 2 2 0 0 -2 -2 90 91 92 93 94 95 96 97 98 99 Target FFR Best rule (GDP deflator) 00 01 02 03 04 05 06 07 08 09 10 Baseline Best rule (GDP deflator), adjusted assumptions The best-fitting rule for the 1990-2002 as well as full sample is the nowcast rule/GDP deflator/CBO output gap; the fit with the Federal Funds rate increases somewhat when assumptions are optimized to fit the full sample. Interest rate [in %] Figure 12: Best-fitting rule for the 2002-2007 sample 10 10 8 8 6 6 4 4 2 2 0 0 -2 -2 90 91 92 93 94 95 96 97 98 Target Federal Funds rate Best rule (PCE core) 99 00 01 02 03 04 05 06 07 08 09 10 Baseline Best rule (PCE core), adjusted assumptions The best-fitting rule for the 2002-2007 sample is the contemporaneous rule/PCE core/ CBO output gap, although this rule is not statistically different from the bestfitting rule over the full sample (see prior figure), suggesting the Federal Reserve did not significantly change its reaction function in 2002. Root mean squared errors are larger on average than over the 1990-2002 sample even when assumptions are optimized to fit the 2002-2007 sample. xii