The Central Bank of Brazil’s time-varying Taylor rule

Carvalho, Filipe Gropelli; Muinhos, Marcelo Kfoury

doi:10.5935/0034-7140.20220022

Filipe Gropelli Carvalho ✉ filipe.carvalho@schroders.com ✉ marcelo.kfoury@fgv.br

Johns Hopkins University School of Advanced International Studies and Schroders Investment Management. Washington, DC, United States

https://orcid.org/0009-0006-1693-3595

Marcelo Kfoury Muinhos ✉ filipe.carvalho@schroders.com ✉ marcelo.kfoury@fgv.br

Escola de Economia de São Paulo (FGV EESP). Rua Itapeva, 474, Bela Vista, São Paulo, SP, CEP 01332-000, Brasil

https://orcid.org/0000-0002-0441-4249

✉ filipe.carvalho@schroders.com ✉ marcelo.kfoury@fgv.br

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Figure 1
Model 1 - Estimating β as only time-varying parameter (plus/minus two standard errors)

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Figure 2
Model 2 - Estimating β and γ as time-varying parameters: β plus/minus two standard errors

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Figure 3
Model 2 - Estimating β and γ as time-varying parameter: γ plus/minus two standard errors

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Figure 4
Model 3 - Estimating β as time-varying parameter with fixed

\bar{rr}

: β plus/minus two standard errors

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Figure 5
Comparison of models 1-3

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Figure 6
Implicit inflation target and official inflation target (plus/minus two standard errors, %)

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Figure 7
Implicit inflation target deviation from official target (percentage points), and 12month expected inflation rate (RHS, %)

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Figure 8
Residuals of instrumental variable estimation for E[π_t]

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Figure 9
Residuals of instrumental variable estimation for γ_t

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Table 1
OLS estimation of initial Taylor rule parameters

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Table 2
Granger causality test between the implicit target deviation from the official target, and inflation expectations

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Table 3
Descriptive statistics of implicit targets by BCB governor

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Table 4
Unit root Test statistics

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Table 5
Unit root with breakpoint test statistics

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Table 6
Model 1 - Estimating β as only time-varying parameter

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Table 7
Model 2 - Estimating β and γ as time-varying parameter

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Table 8
Model 3 - Estimating β as time-varying parameter with fixed rr

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Table 9
Model 4 - Implicit inflation target

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(2)

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(6b)

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(7c)

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Coefficient	Estimated Value	Standard Error	t-statistic	p-value
ρ	0.943704	0.025789	36.59342	0
rr	0.05334	0.01847	2.887984	0.0054
β	3.847894	2.035896	1.890025	0.0636
γ	2.374615	1.052556	2.256047	0.0277
D2003Q4	-0.021782	0.006213	-3.505984	0.0009
^DGFC(2009Q1)	-0.017385	0.006004	-2.895674	0.0053
D2011Q3	-0.013989	0.005958	-2.348115	0.0222
D2012Q2	-0.015182	0.005987	-2.535733	0.0138
^DTruckerStrike(Q32018)	0.016710	0.006044	2.764695	0.0076
R² = 0.968325		Adjusted R² = 0.964101

	F-Statistic	p-value
Implicit target deviation from official target does not Granger cause inflation expectations	3.26156	0.0177

	Meirelles	Tombini	Goldfajn	Campos Neto
Number of quarters in sample period (n)	30	22	10	7
Below bottom band	0 (0%)	0 (0%)	0 (0%)	0 (0%)
Between center and bottom band	24	2	5	2
	(80%)	(9%)	(50%)	(29%)
Between center and upper band	6	17	5	5
	(20%)	(77%)	(50%)	(71%)
Above upper band	0	3	0	0
	(0%)	(14%)	(0%)	(0%)
Mean squared deviation from official inflation target	1.168262	1.957557	0.078214	0.16439

\begin{aligned} rr t & = (1 - ρ) (\bar{rr} + (β - 1) (E [π_{t, n} | Ω_{t}] - π^{*}) + γ {\tilde{y}}_{t}) + ρ rr t - 1 + ε_{t}, \\ ε_{t} \sim i.i.d. N (0, σ^{2}); \end{aligned}

\begin{aligned} rr t & = (1 - ρ) (\bar{rr} + (β - 1) (E [π_{t, n}] | Ω_{t} - π^{*}) + γ {\tilde{y}}_{t}) + ρ rr t - 1 + ε_{t}, \\ ε_{t} \sim i.i.d. N (0, σ^{2}); \end{aligned}

\begin{aligned} r_{t} & = (1 - ρ) (\bar{rr} + (β - 1) (E [π_{t, n}] | Ω_{t} - π_{t}^{*}) + y {\tilde{y}}_{t}) + ρ r_{t - 1} + ε_{t}, \\ ε_{t} \sim i.i.d. N (0, σ^{2}), \end{aligned}

{rr}_{t} = (1 - ρ) (\bar{rr} + (β - 1) (E [π_{t, n} | Ω_{t}] - π^{*}) + y {\tilde{γ}}_{t}) + ρ {rr}_{t - 1} + ν_{1} σ_{e} υ_{1 t}^{*} + ν_{2} σ_{e} υ_{2 t}^{*} + Ω_{t}, Ω_{t} \sim N (0, (1 - ν_{1}^{2} - ν_{2}^{2}) σ_{e, t}^{2}),

Variable	ADF	PP	KPSS
Real interest rate (rr_t)	-0.3151	-2.6802	0.1206^*
Inflation expectations (E[π])	-4.6619^***	-4.4178^***	0.1473^**
Inflation target (π^∗)	-3.9597^**	-4.0871^**	0.0544
Output gap (γ)	-1.7250	-3.5214^**	0.2153^**
US Dollar to Brazilian Reais (dlfx)	-6.3676^***	-6.1147^***	0.0492
WTI oil prices (dloil)	-7.2571^***	-7.3557^***	0.0585
Monthly industrial production (dlpim)	-8.7574^***	-8.6235^***	0.0751

Variable	ADF
Real interest rate (rr_t)	-5.7029^ p < 0.05
Output gap (γ)	-5.8752^* ***** p < 0.01

Coefficient	Estimated Value	Standard Error	z-Statistic	p-value
ρ	-4.080052	0.250062	-16.31618	0.0000
$\bar{rr}$	0.069156	0.005964	11.59481	0.0000
γ	0.520082	0.217832	2.387535	0.0170
v ₁	-670.1163	2026.426	-0.330689	0.7409
v ₂	-702.9792	1045.290	-0.672521	0.5013
TVP	Final State	Root MSE	z-Statistic	p-value
β	2.412928	0.905056	2.666053	0.0077
Log-likelihood	38.36762	Akaine I.C.		-0.967177
Parameters	5	Schwarz I.C.		-0.805286
Diffuse priors	1	Hannan-Quinn I.C.		-0.902949

Coefficient	Estimated Value	Standard Error	z-Statistic	p-value
ρ	0.000528	0.000112	4.699920	0.0000
$\bar{rr}$	0.067951	0.003477	19.54500	0.0000
v ₁	-262.2820	337.8877	-0.776240	0.4376
v ₂	-33.63888	156.2058	-0.215350	0.8295
TVP	Final State	Root MSE	z-Statistic	p-value
β	0.870547	0.558089	1.559870	0.1188
γ	0.898746	0.256183	3.508214	0.0005
Log-likelihood	139.3330	Akaine I.C.		-3.922694
Parameters	4	Schwarz I.C.		-3.793181
Diffuse priors	2	Hannan-Quinn I.C.		-3.871312

Coefficient	Estimated Value	Standard Error	z-Statistic	p-value
ρ	-3.710086	0.212645	-17.44736	0.0000
γ	0.398050	0.305797	1.301681	0.1930
v ₁	-954.5989	2589.041	-0.368708	0.7123
v ₂	-657.2078	1216.875	-0.540078	0.5891
TVP	Final State	Root MSE	z-Statistic	p-value
β	1.538786	0.523789	2.937801	0.0033
Log-likelihood	32.34217	Akaine I.C.		-0.821512
Parameters	4	Schwarz I.C.		-0.691999
Diffuse priors	1	Hannan-Quinn I.C.		-0.770130

Coefficient	Estimated Value	Standard Error	z-Statistic	p-value
ρ	0.742983	0.053893	13.78630	0.0000
$\bar{rr}$	0.051069	54259861	9.41E-10	1.0000
β	3.622114	0.524541	6.905302	0.0000
γ	0.540605	0.278322	1.942370	0.0521
v ₁	-239.9813	71.22085	-3.369537	0.0008
v ₂	-35.39408	25.51706	-1.387075	0.1654
TVP	Final State	Root MSE	z-Statistic	p-value
π∗	0.045285	0.004865	9.309075	0.0000
Log-likelihood	228.8570	Akaine I.C.		-6.459622
Parameters	6	Schwarz I.C.		-6.265352
Diffuse priors	1	Hannan-Quinn I.C.		-6.382549

[\begin{matrix} υ_{1, t | t - 1} \\ υ_{2, t | t - 1} \end{matrix}] = Ω_{t | t - 1}^{1 / 2} [\begin{matrix} υ_{1, t}^{*} \\ υ_{2, t}^{*} \end{matrix}], [\begin{matrix} υ_{1, t}^{*} \\ υ_{2, t}^{*} \end{matrix}] \sim i . i . d . N ([\begin{matrix} 0 \\ 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]),

[\begin{matrix} υ_{t}^{*} \\ e_{t} \end{matrix}] = [\begin{matrix} I_{2} & 0_{2} \\ ρ^{'} σ_{e, t} & \sqrt{(1 - ρ^{'} ρ)} σ_{e, t} \end{matrix}] [\begin{matrix} ϵ_{t} \\ ω_{t} \end{matrix}], [\begin{matrix} ϵ_{t} \\ ω_{t} \end{matrix}] \sim i . i . d . N ([\begin{matrix} 0_{2} \\ 0 \end{matrix}], [\begin{matrix} I_{2} & 0_{2} \\ 0_{2}^{'} & 1 \end{matrix}]),

[1] ✉ filipe.carvalho@schroders.com ✉ marcelo.kfoury@fgv.br

Brasil

Brasil

The Central Bank of Brazil’s time-varying Taylor rule

Abstract