3.6 Appendix

3.6.1 Data description

Interest rates:

Australia: The key policy rate used in the regressions is the monthly average inter-bank overnight rate (the cash rate).

Japan: The policy rate used is the target uncollateralized middle monthly average rate.

New Zealand: The policy rate used is the official overnight target rate (the cash rate).

Norway: As a policy rate we have used the overnight market lending rate since March 1987. Prior to this date, the discount rate has been used as a policy rate.

Sweden: Since June 1994, the policy rate used is the repo rate. Prior to this date we have used the discount rate.

Switzerland: Since January 2000, the policy rate used is the target for the 3 month Libor rate on the CHF. Prior to 2000, we have used the discount rate.

USA: The interest rate used is the monthly average effective federal funds rate.

Canada: Since May 2001, the key policy rate used is the Bank of Canada target for the overnight rate. Prior to May 2001, the Bank rate has been used in the regressions.

All interest rates are reported for the month following the release of the inflation and real GDP forecasts within the same quarter in order to avoid a possible endogeneity problem.

Table 3.1: Summary statistics, OECD forecasts

DEPENDENT AND EXPLANATORY VARIABLES Obs. Mean Std. deviation Min Max

Australia

Policy interest rate (%) 138 8.711 4.358 3.00 19.39

Inflation forecasts (%) 138 5.228 3.618 -0.34 14.23

Real GDP growth forecasts (%) 138 3.216 1.978 -3.07 8.36

Output gap change forecasts (%) 138 2.73e-9 1.776 -5.84 5.03

Japan

Policy interest rate (%) 138 2.529 2.421 0.10 9.00

Inflation forecasts (%) 138 1.587 2.544 -2.30 10.50

Real GDP growth forecasts (%) 138 2.506 2.730 -8.69 9.27

Output gap change forecasts (%) 138 1.20e-8 1.927 -8.56 4.74

New Zealand

Policy interest rate (%) 138 10.133 5.075 2.50 28.20

Inflation forecasts (%) 138 6.308 5.780 -0.51 18.93

Real GDP growth forecasts (%) 138 2.197 3.319 -12.89 14.72

Output gap change forecasts (%) 138 1.04e-9 2.811 -11.82 11.19

Norway

Policy interest rate (%) 138 7.609 2.569 2.25 13.80

Inflation forecasts (%) 138 4.562 3.424 -1.30 14.63

Real GDP growth forecasts (%) 138 2.846 2.243 -2.39 8.29

Output gap change forecasts (%) 138 3.00e-9 1.757 -4.19 4.87

Sweden

Policy interest rate (%) 138 6.278 3.188 0.25 12.00

Inflation forecasts (%) 138 4.691 4.116 -1.12 14.73

Real GDP growth forecasts (%) 138 2.013 2.382 -6.61 5.76

Output gap change forecasts (%) 138 9.47e-9 1.905 -7.18 4.56

Switzerland

Policy interest rate (%) 138 2.728 1.908 0.25 7.00

Inflation forecasts (%) 138 2.110 1.850 -0.97 7.16

Real GDP growth forecasts (%) 138 1.734 1.797 -3.32 6.17

Output gap change forecasts (%) 138 4.31e-7 1.547 -4.76 3.91

USA

Policy interest rate (%) 138 6.031 3.834 0.12 19.10

Inflation forecasts (%) 138 4.085 2.900 -1.60 14.41

Real GDP growth forecasts (%) 138 2.868 2.211 -4.11 8.48

Output gap change forecasts (%) 138 -1.21e-9 1.748 -5.08 4.93

Canada

Policy interest rate (%) 138 7.136 4.248 0.25 20.04

Inflation forecasts (%) 138 4.030 3.169 -0.89 12.68

Real GDP growth forecasts (%) 138 2.683 2.276 -3.71 6.55

Output gap change forecasts (%) 138 4.90e-9 1.819 -5.98 3.39

Note: The policy interest rates are taken from the Central Banks’ official websites and Datastream. The inflation and real GDP growth forecasts are performed by the OECD and are downloaded from Datastream. The forecasts for the change in the output gap are computed by the authors using an H-P filter.

3.6. Appendix 117

3.6.2 Estimation tables, OECD forecasts

Table 3.2: Semiparametric model with interactions, output growth

Australia Japan New Zealand Norway Sweden Switzerland USA Canada

constant 0.087 0.025 0.101 0.076 0.063 0.027 0.060 0.071

(2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)***

sπ(Etπt+1) 1.36 1.00 3.22 1.00 1.00 1.00 1.00 2.24

(0.46) (6e-5)*** (2e-16)*** (6e-3)*** (0.14) (1e-8)*** (0.066)* (3e-6)***

sx(Etxt+1) 1.22 1.00 1.00 1.00 1.00 2.56 1.00 1.00

(0.60) (0.96) (0.23) (0.38) (0.020)** (0.87) (0.068)* (0.016)**

sπ,x(Etπt+1, Etxt+1) 7.09 12.37 2.00 8.10 19.84 4.12 13.25 11.79

(8e-4)*** (3e-8)*** (0.99) (8e-5)*** (2e-8)*** (0.43) (4e-4)*** (9e-4)***

Obs. 138 138 138 138 138 138 138 138

Adj.R2 0.648 0.826 0.552 0.395 0.793 0.610 0.716 0.799

Dev. Exp. 67.3% 84.4% 56.6% 43.9% 82.6% 63.2% 74.7% 82.1%

AIC -605.0 -860.4 -539.7 -675.6 -754.1 -820.7 -664.8 -684.5

Note: ***, ** and * denote p-values smaller than 0.01, 0.05 and 0.10 respectively.

Table 3.3: Semiparametric model with no interactions, output growth

Australia Japan New Zealand Norway Sweden Switzerland USA Canada

constant 0.087 0.025 0.101 0.076 0.063 0.027 0.060 0.071

(2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** 2e-16)***

sπ(Etπt+1) 3.74 4.84 3.22 2.53 2.53 1.23 1.00 3.56

(2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (1e-07)*** (2e-16)*** (2e-16)*** (2e-16)***

sx(Etxt+1) 2.36 1.00 1.00 1.21 1.74 1.00 1.84 4.51

(0.02)** (2e-4)*** (0.24) (0.54) (0.24) (0.79) (8e-3)*** (4e-3)***

Obs. 138 138 138 138 138 138 138 138

Adj.R2 0.624 0.77 0.552 0.234 0.688 0.578 0.606 0.757

Dev. Exp. 64.1% 77.9% 56.6% 25.5% 69.7% 58.5% 61.5% 77.1%

AIC -599.2 -829.3 -534.7 -649.1 -712.9 -814.9 -631.4 -664.6

Note: ***, ** and * denote p-values smaller than 0.01, 0.05 and 0.10 respectively.

Table 3.4: Parametric Taylor rule with interactions, output growth

Australia Japan New Zealand Norway Sweden Switzerland USA Canada

constant 0.035 0.009 0.061 0.058 0.033 0.009 0.010 0.016

(2e-3)*** (4e-8)*** (2e-15)*** (2e-16)*** (2e-16)*** (2e-5)*** (0.056)* (2e-3)***

Etπt+1 0.87 0.68 0.62 0.42 0.59 0.84 1.06 1.18

(4e-9)*** (2e-11)*** (2e-16)*** (9e-6)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)***

Etxt+1 0.264 0.206 0.017 0.167 -0.004 0.119 0.380 0.433

(0.361) (4e-5)*** (0.93) (0.31) (0.97) (0.21) (0.013)** (5e-3)***

Etπt+1×Etxt+1 -1.40 0.29 0.92 -4.63 3.10 -4.91 -2.92 -4.16

(0.71) (0.91) (0.59) (0.098)* (0.12) (0.085)* (0.30) (0.111)

Obs. 138 138 138 138 138 138 138 138

Adj.R2 0.438 0.708 0.503 0.167 0.626 0.583 0.595 0.658

AIC -546.8 -796.1 -518.5 -638.3 -689.3 -815.7 -627.3 -622.5

Note: ***, ** and * denote p-values smaller than 0.01, 0.05 and 0.10 respectively.

Table 3.5: Parametric Taylor rule with no interactions, output growth

Australia Japan New Zealand Norway Sweden Switzerland USA Canada

constant 0.038 0.009 0.059 0.064 0.032 0.027 0.060 0.071

(1e-6)*** (4e-8)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)***

Etπt+1 0.83 0.69 0.64 0.30 0.63 1.23 1.00 3.56

(2e-16)*** (2e-16)*** (2e-16)*** (8e-7)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)***

Etxt+1 0.17 0.21 0.10 -0.06 0.09 1.00 1.84 4.51

(0.24) (2e-4)*** (0.32) (0.49) (0.24) (0.79) (8e-3)*** (4e-3)***

Obs. 138 138 138 138 138 138 138 138

Adj.R2 0.442 0.703 0.494 0.156 0.622 0.578 0.606 0.757

AIC -548.6 -798.1 -520.2 -637.5 -688.7 -814.9 -631.4 -664.6

Note: ***, ** and * denote p-values smaller than 0.01, 0.05 and 0.10 respectively.

Table 3.6: Semiparametric model with interactions, speed limit policy

Australia Japan New Zealand Norway Sweden Switzerland USA Canada

constant 0.087 0.025 0.101 0.076 0.063 0.027 0.060 0.071

(2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)***

sπ(Etπt+1) 1.33 1.00 2.40 2.57 2.57 1.00 1.00 1.91

(0.604) (1e-7)*** (0.028)** (6e-7)*** (2e-16)*** (2e-16)*** (0.014)** (1e-4)***

sx(Etxt+1) 1.480 1.07 1.00 1.57 2.20 1.00 1.00 1.00

(0.56) (0.77) (0.71) (0.17) (0.029)** (0.95) (0.30) (0.031)**

sπ,x(Etπt+1, Etxt+1) 6.86 7.51 7.76 2.00 2.00 2.00 16.59 13.58

(2e-9)*** (4e-11)*** (3e-3)*** (0.12) (0.36) (0.16) (5e-6)*** (7e-7)***

Obs. 138 138 138 138 138 138 138 138

Adj.R2 0.644 0.758 0.598 0.253 0.704 0.579 0.701 0.797

Dev. Exp. 66.9% 77.5% 63.1% 27.5% 71.4% 58.6% 74.1% 82.1%

AIC -603.3226 -819.2668 -543.4079 -652.1809 -719.7 -815.0071 -654.9837 -681.833

Note: ***, ** and * denote p-values smaller than 0.01, 0.05 and 0.10 respectively.

Table 3.7: Semiparametric model with no interactions, speed limit policy

Australia Japan New Zealand Norway Sweden Switzerland USA Canada

constant 0.087 0.025 0.101 0.076 0.063 0.027 0.060 0.071

(2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)***

sπ(Etπt+1) 3.66 3.86 3.22 2.57 2.57 1.02 1.72 3.63

(2e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (6e-7)*** (2e-16)*** (2e-16)*** (2e-16)***

sx(Etxt+1) 2.99 1.00 1.00 1.57 2.20 1.00 2.84 1.00

(0.083)* (0.48) (0.96) (0.11) (0.025)** (0.97) (0.33) (9e-3)***

Obs. 138 138 138 138 138 138 138 138

Adj.R2 0.616 0.74 0.547 0.253 0.704 0.576 0.594 0.739

Dev. Exp. 63.5% 74.9% 56.1% 27.5% 71.4% 58.3% 60.7% 74.8%

AIC -596.0 -813.6 -533.3 -652.2 -719.7 -814.6 -625.4 -658.0

Note: ***, ** and * denote p-values smaller than 0.01, 0.05 and 0.10 respectively.

3.6. Appendix 119 Table 3.8: Parametric Taylor rule with interactions, speed limit policy

Australia Japan New Zealand Norway Sweden Switzerland USA Canada

constant 0.044 0.013 0.062 0.062 0.034 0.011 0.018 0.027

(2e-15)*** (4e-15)*** (2e-16)*** (2e-16)*** (2e-16)*** (4e-10)*** (6e-6)*** (2e-12)***

Etπt+1 0.81 0.801 0.617 0.310 0.615 0.77 1.03 1.10

(2e-16)*** (2e-16)*** (2e-16)*** (2e-6)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)***

Etxt+1 0.456 -0.041 0.230 -0.376 0.026 0.092 0.019 0.395

(0.18) (0.53) (0.29) (0.078)* (0.82) (0.38) (0.93) (0.045)**

Etπt+1×Etxt+1 -4.5 6.4 -2.3 4.1 0.32 -3.7 2.5 -1.3

(0.29) (0.047)** (0.24) (0.20) (0.90) (0.23) (0.49) (0.66)

Obs. 138 138 138 138 138 138 138 138

Adj.R2 0.440 0.663 0.492 0.168 0.616 0.578 0.574 0.655

AIC -547.2 -779.4 -518.6 -638.4 -685.5 -814.1 -620.4 -621.2

Note: ***, ** and * denote p-values smaller than 0.01, 0.05 and 0.10 respectively.

Table 3.9: Parametric Taylor rule with no interactions, speed limit policy

Australia Japan New Zealand Norway Sweden Switzerland USA Canada

constant 0.045 0.013 0.062 0.063 0.034 0.011 0.019 0.027

(2e-15)*** (9e-16)*** (2e-16)*** (2e-16)*** (2e-16)*** (7e-10)*** (9e-7)*** (2e-12)***

Etπt+1 0.815 0.773 0.620 0.291 0.613 0.787 1.01 1.10

(2e-16)*** (2e-16)*** (2e-16)*** (3e-6)*** (2e-16)*** (2e-16)*** (2e-16)*** (2e-16)***

Etxt+1 0.141 -0.010 0.011 -0.15 0.035 -0.002 0.132 0.327

(0.38) (0.87) (0.92) (0.21) (0.70) (0.98) (0.28) (6e-3)***

Obs. 138 138 138 138 138 138 138 138

Adj.R2 0.439 0.491 0.494 0.163 0.619 0.576 0.576 0.657

AIC -548.0 -777.3 -519.2 -638.7 -687.5 -814.6 -621.9 -623.0

Note: ***, ** and * denote p-values smaller than 0.01, 0.05 and 0.10 respectively.

3.6.3 Nonlinearity tests

Table 3.10: Nonlinearity tests, output growth

Australia Japan New Zealand Norway Sweden Switzerland USA Canada

τ1 0.044** 0.006*** 0.839 0.008*** 0.001*** 0.010** 0.012** 0.035**

ASP vs. SP τ2 0.006*** 0.044** 0.457 0.009*** 0.001*** 0.012** 0.012** 0.015**

τ3 0.040** 0.007*** 0.830 0.008*** 0.001*** 0.010** 0.012** 0.057*

τ1 0.037** 0.001*** 0.035** 0.010*** 0.001*** 0.092* 0.033** 0.001***

ASP vs. AP τ2 0.04** 0.001*** 0.026** 0.008*** 0.001*** 0.047** 0.031** 0.001***

τ3 0.044** 0.001*** 0.040** 0.009*** 0.001*** 0.059* 0.030** 0.001***

τ1 0.054* 0.001*** 0.038** 0.010*** 0.001*** 0.041** 0.030** 0.001***

ASP vs. P τ2 0.044** 0.001*** 0.028** 0.013** 0.001*** 0.064* 0.028** 0.001***

τ3 0.056* 0.001*** 0.050** 0.012** 0.001*** 0.033** 0.030** 0.001***

τ1 0.006*** 0.001*** 0.013** 0.062* 0.001*** 0.351 0.406 0.001***

SP vs. P τ2 0.008*** 0.001*** 0.012** 0.057* 0.001*** 0.349 0.393 0.001***

τ3 0.006*** 0.001*** 0.014** 0.069* 0.001*** 0.351 0.407 0.001***

Note: ***, ** and * denote p-values smaller than 0.01, 0.05 and 0.10 respectively.

Table 3.11: Nonlinearity tests, speed limit policy

Australia Japan New Zealand Norway Sweden Switzerland USA Canada

τ1 0.030** 0.029** 0.218 0.863 0.512 0.075* 0.002*** 0.001***

ASP vs. SP τ2 0.033** 0.038** 0.249 0.360 0.702 0.097* 0.002*** 0.001***

τ3 0.031** 0.027** 0.207 0.858 0.502 0.072* 0.001*** 0.001***

τ1 0.029** 0.004*** 0.126 0.034** 0.002*** 0.421 0.020** 0.001***

ASP vs. AP τ2 0.028** 0.009*** 0.132 0.041** 0.007*** 0.646 0.021** 0.001***

τ3 0.033** 0.002*** 0.131 0.031** 0.003*** 0.048** 0.018** 0.001***

τ1 0.051* 0.001*** 0.117 0.037** 0.007*** 0.305 0.021** 0.001***

ASP vs. P τ2 0.043** 0.001*** 0.122 0.025** 0.008*** 0.330 0.025** 0.001***

τ3 0.055* 0.001*** 0.118 0.039** 0.005*** 0.297 0.020** 0.001***

τ1 0.012** 0.001*** 0.083* 0.011** 0.001*** 0.341 0.309 0.001***

SP vs. P τ2 0.012** 0.001*** 0.078* 0.010*** 0.001*** 0.34 0.322 0.001***

τ3 0.008*** 0.002*** 0.092* 0.010*** 0.001*** 0.341 0.304 0.001***

Note: ***, ** and * denote p-values smaller than 0.01, 0.05 and 0.10 respectively.

3.6. Appendix 121

3.6.4 Figures

Australia Bivariate model

Figure 3.1: Estimated inflation function of the Policy Rate of Australia

Figure 3.2: Estimated output growth function of the Policy Rate of Australia

Figure 3.3: Estimated surface of the in-teraction term for the Policy Rate of Aus-tralia

Figure 3.4: Estimated surface with asymptotic confidence surface of the in-teraction term for the Policy Rate of Aus-tralia

Japan Bivariate model

Figure 3.5: Estimated inflation function of the Policy Rate of Japan

Figure 3.6: Estimated output growth function of the Policy Rate of Japan

Figure 3.7: Estimated surface of the inter-action term for the Policy Rate of Japan

Figure 3.8: Estimated surface with asymptotic confidence surface of the in-teraction term for the Policy Rate of Japan

3.6. Appendix 123 New Zealand Bivariate model

Figure 3.9: Estimated inflation function of the Policy Rate of New Zealand

Figure 3.10: Estimated output growth function of the Policy Rate of New Zealand

Figure 3.11: Estimated surface of the in-teraction term for the Policy Rate of New Zealand

Figure 3.12: Estimated surface with asymptotic confidence surface of the in-teraction term for the Policy Rate of New Zealand

Norway Bivariate model

Figure 3.13: Estimated inflation function of the Policy Rate of Norway

Figure 3.14: Estimated output growth function of the Policy Rate of Norway

Figure 3.15: Estimated surface of the in-teraction term for the Policy Rate of Nor-way

Figure 3.16: Estimated surface with asymptotic confidence surface of the in-teraction term for the Policy Rate of Nor-way

3.6. Appendix 125 Sweden Bivariate model

Figure 3.17: Estimated inflation function of the Policy Rate of Sweden

Figure 3.18: Estimated output growth function of the Policy Rate of Sweden

Figure 3.19: Estimated surface of the in-teraction term for the Policy Rate of Swe-den

Figure 3.20: Estimated surface with asymptotic confidence surface of the in-teraction term for the Policy Rate of Swe-den

Switzerland Bivariate model

Figure 3.21: Estimated inflation function of the Policy Rate of Switzerland

Figure 3.22: Estimated output growth function of the Policy Rate of Switzerland

Figure 3.23: Estimated surface of the interaction term for the Policy Rate of Switzerland

Figure 3.24: Estimated surface with asymptotic confidence surface of the in-teraction term for the Policy Rate of Switzerland

3.6. Appendix 127 USA Bivariate model

Figure 3.25: Estimated inflation function of the Policy Rate of USA

Figure 3.26: Estimated output growth function of the Policy Rate of USA

Figure 3.27: Estimated surface of the in-teraction term for the Policy Rate of USA

Figure 3.28: Estimated surface with asymptotic confidence surface of the in-teraction term for the Policy Rate of USA

Canada Bivariate model

Figure 3.29: Estimated inflation function of the Policy Rate of Canada

Figure 3.30: Estimated output growth function of the Policy Rate of Canada

Figure 3.31: Estimated surface of the interaction term for the Policy Rate of Canada

Figure 3.32: Estimated surface with asymptotic confidence surface of the in-teraction term for the Policy Rate of Canada

Conclusion

In the first chapter, we provide the necessary conditions to achieve local robust-ness for valid and invalid moments in a GMM setting. Based on this result and on Moment Selection Criteria developed in [Andrews, 1999], we propose a robust way of consistently selecting the set of moments that maximizes the number of orthogonality conditions that are asymptotically zero, denoted by RMSC.

We show that RMSC is a consistent procedure when the underlying distribution holds true. Moreover, we derive the post-selection properties of RMSC. Monte-Carlo experiments show a very unstable behavior of the classical procedures in the selection of the set of moments that maximizes the number of orthogonality conditions that are asymptotically zero even for very small contaminations. In contrast to this finding the robust procedures display a very steady performance in all settings. The procedure is also applied in a real data set.

Based on the construction of RGMM from Chapter 1 and work by

[Andrews and Lu, 2001], we derive Robust Model and Moment Selection Criteria (RMMSC). RMMSC selects the set of parameters and moments that maximize the number of orthogonality conditions that are asymptotically zero.

We apply the methodology in the IV setting. We make connections between higher-order robustness and the RGMM used in this paper. We show that RGMM is second-order robust for consistent moments but it is not, in general, for inconsistent moments. More specifically, in the linear IV context the RGMM estimator we provide is second-order robust for both valid and invalid sets of moments.

We study the finite-sample behavior of the procedures through Monte-Carlo simu-lations. RMMSC performs equally well as MMSC with no contamination. However, RMMSC outperforms MMSC in the presence of contamination.

Possible directions for further research are as follows. Since Hansen’s statistic is known to perform poorly in finite samples, a more careful investigation of the ro-bustness properties of MSC beyond the standard asymptotic framework would be useful. For example, by means of Generalized Likelihood Estimators, see

[Hansen et al., 1996] and [Imbens, 2002]. The existing estimators based on the lat-ter do not have a bounded influence function in general. Hence, they are not locally robust. The main challenge, in our opinion, is to have a unified theory of local robustness for this class of estimators. There exist some robust counterparts of the Generalized Exponential Tilting estimator by [Lˆo and Ronchetti, 2012]. To our knowledge, there is neither robust counterpart of the very well-known Generalized Empirical Likelihood estimator nor for the Continuous Updating Estimator.

The third chapter contributes to the literature by proposing two new semipara-metric reaction functions that include the parasemipara-metric Taylor rule as a special case.

From a policy-making perspective, the new semiparametric policy rules are easy to implement and relatively simple to communicate to the public. The semiparametric policy rules are in line with asymmetric Central Banks preferences and/or a nonlin-ear structure of the economy. The former permits to track more closely the actual interest rate decision process of Central Banks than their parametric counterpart.

Furthermore, the semiparametric reaction functions are immune to misspecification problems introduced by the presence of a lagged dependent variable as a covariate.

The estimation results show clear evidence for a nonlinear Central Bank responsive-ness to macroeconomic fundamentals for most of the countries studied. The latter is further confirmed by the three statistical tests for nonlinearities that we perform.

The semiparametric reaction function with an interaction term is the preferred spec-ification for all countries except for New Zealand and Switzerland. This evidence is robust to different measures of the economic outlook.

Conclusion 131 Future research should focus on estimating the semiparametric policy rules with structural breaks in time. Given the strong empirical support for the augmented semiparametric specification, we hope that future research will focus on understand-ing more deeply the role of interaction terms and the effect on monetary policy predictability. Moreover, following a bottom-up approach it would be valuable to parameterize the nonlinear specifications in order to further enhance Central Bank communication with the relevant market participants.

Appendices

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