Modelling inflation dynamics: a Bayesian comparison between GARCH and stochastic volatility

Figures & data

Table 1. Summary statistics and unit root tests.

Country	Mean	Median	Std.Dev	Skew	Ex.Kur	JB	ADF	PP
Canada	3.55	2.61	3.50	1.00	1.05	50.55${}^{***}$	−2.80${}^{***}$	−5.92${}^{***}$
France	4.05	2.67	3.72	1.15	0.57	55.22${}^{***}$	−1.81${}^{*}$	−2.88${}^{**}$
Germany	2.61	2.21	1.90	0.80	0.30	26.18${}^{***}$	−2.08${}^{**}$	−4.96${}^{***}$
Italy	5.67	3.97	5.36	1.47	1.76	115.86${}^{***}$	−1.69${}^{*}$	−2.70${}^{*}$
Japan	2.92	1.69	4.57	2.58	12.83	1878.9${}^{***}$	−3.66${}^{***}$	−6.44${}^{***}$
United Kingdom	5.06	3.26	4.97	2.17	5.89	525.56${}^{***}$	−2.35${}^{**}$	−4.35${}^{***}$
United States	3.70	3.19	3.03	0.81	2.92	110.65${}^{***}$	−2.60${}^{***}$	−5.23${}^{***}$
Australia	4.65	3.70	4.37	1.25	2.25	111.34${}^{***}$	−2.71${}^{***}$	−7.69${}^{***}$
Austria	3.23	2.75	2.34	1.09	1.48	68.37${}^{***}$	−2.74${}^{***}$	−6.96${}^{***}$
Belgium	3.48	2.90	3.01	1.28	2.73	137.94${}^{***}$	−2.47${}^{**}$	−5.78${}^{***}$
Denmark	4.31	2.71	4.17	1.35	1.98	98.93${}^{***}$	−3.20${}^{***}$	−6.39${}^{***}$
Finland	4.51	3.31	4.37	1.16	0.93	61.17${}^{***}$	−2.21${}^{**}$	−3.88${}^{***}$
Luxembourg	3.29	2.74	2.83	0.99	0.84	45.40${}^{***}$	−2.46${}^{**}$	−5.43${}^{***}$
Netherlands	3.31	2.55	2.87	0.94	1.04	45.61${}^{***}$	−2.82${}^{***}$	−7.13${}^{***}$
New Zealand	5.39	3.47	5.43	1.42	2.73	153.05${}^{***}$	−2.64${}^{***}$	−4.96${}^{***}$
Norway	4.46	3.61	3.79	1.05	2.97	130.89${}^{***}$	−2.87${}^{***}$	−8.02${}^{***}$
Sweden	4.26	2.96	4.17	1.11	1.21	62.94${}^{***}$	−2.89${}^{***}$	−6.82${}^{***}$
Switzerland	2.45	1.77	2.62	1.18	2.99	143.19${}^{***}$	−2.75${}^{***}$	−4.71${}^{***}$

Notes: ${}^{*},{}^{**},$ and ${}^{***}$ indicate the significance level of 10%, 5%, and 1%, respectively. Due to space constraints, we only report: (1) an ADF test in the absence of drift and trend; (2) a PP test with intercept. The period spans from 1961Q1 to 2018Q4 for all countries except for Denmark (1967Q1 to 2018Q4). Note also that here we display an ‘Excess Kurtosis’ (Ex.Kur), which is simply a ‘Kurtosis-3’.

Source: Author.

Table 2. Log marginal likelihood of two classes of volatility models for 18 rich OECD countries’ inflation. The numbers in parentheses are numerical standard errors.

Type	GARCH	SV	GARCH-GJR	SV-L	GARCH-M	SV-M
Canada	−569.8	−562.5	−563.9	−556.4	−555.8	−516.8
	(0.09)	(0.01)	(0.15)	(0.02)	(0.02)	(0.03)
France	−543.8	−534.3	−551.5	−535.4	−479.0	−418.8
	(0.24)	(0.01)	(0.07)	(0.02)	(0.09)	(0.04)
Germany	−440.6	−447.1	−440.2	−444.4	−431.0	−354.6
	(0.29)	(0.01)	(0.13)	(0.01)	(0.08)	(0.03)
Italy	−599.2	−587.6	−606.8	−588.4	−543.9	−450.7
	(0.31)	(0.01)	(0.15)	(0.02)	(0.06)	(0.03)
Japan	−613.7	−597.9	−611.7	−599.2	−585.0	−524.4
	(0.11)	(0.04)	(0.18)	(0.04)	(0.06)	(0.02)
United Kingdom	−583.0	−565.9	−597.1	−567.0	−535.3	−476.9
	(0.36)	(0.01)	(0.36)	(0.03)	(0.07)	(0.04)
United States	−513.8	−520.0	−511.7	−517.1	−515.0	−482.1
	(0.11)	(0.03)	(0.32)	(0.02)	(0.05)	(0.03)
Australia	−578.3	−564.4	−573.5	−563.3	−551.7	−508.4
	(0.10)	(0.01)	(0.15)	(0.02)	(0.03)	(0.03)
Austria	−496.7	−486.0	−495.8	−486.6	−474.1	−435.8
	(0.21)	(0.01)	(0.25)	(0.02)	(0.05)	(0.03)
Belgium	−527.1	−536.4	−527.9	−533.6	−522.4	−483.9
	(0.13)	(0.02)	(0.10)	(0.01)	(0.05)	(0.04)
Denmark	−524.7	−483.5	−530.2	−485.0	−461.8	−435.2
	(0.29)	(0.01)	(0.21)	(0.01)	(0.04)	(0.02)
Finland	−610.8	−605.1	−608.1	−604.2	−551.4	−490.8
	(0.14)	(0.01)	(0.17)	(0.03)	(0.08)	(0.03)
Luxembourg	−523.0	−521.5	−523.2	−520.6	−517.8	−465.1
	(0.07)	(0.03)	(0.16)	(0.02)	(0.03)	(0.05)
Netherlands	−531.2	−518.7	−534.6	−519.9	−511.7	−485.3
	(0.12)	(0.01)	(0.10)	(0.01)	(0.05)	(0.03)
New Zealand	−658.7	−645.2	−653.1	−642.5	−637.4	−562.0
	(0.44)	(0.01)	(0.31)	(0.02)	(0.04)	(0.03)
Norway	−630.9	−598.9	−628.4	−599.1	−599.3	−555.7
	(0.05)	(0.02)	(0.10)	(0.02)	(0.07)	(0.02)
Sweden	−641.3	−624.2	−639.0	−619.7	−610.0	−533.9
	(0.18)	(0.01)	(0.17)	(0.02)	(0.04)	(0.01)
Switzerland	−502.3	−513.6	−501.6	−511.4	−490.2	−413.6
	(0.38)	(0.02)	(0.26)	(0.02)	(0.07)	(0.03)

Source: Author.

Table 3. Bayesian estimation for the GARCH models: Estimated posterior means (posterior standard error in parentheses).

Countries	Models	α	β	δ	γ	θ	λ
Canada	GARCH	2.78	1.35	0.42	0.45
		(0.02)	(0.46)	(0.09)	(0.09)
	GARCH-GJR	2.71	1.91	0.54	0.37	−0.39
		(0.05)	(0.49)	(0.10)	(0.09)	(0.11)
	GARCH-M	1.54	0.41	0.31	0.65		0.09
		(0.17)	(0.19)	(0.06)	(0.06)		(0.02)
France	GARCH	3.14	0.51	0.38	0.54
		(0.06)	(0.26)	(0.09)	(0.10)
	GARCH-GJR	3.16	0.56	0.40	0.53	−0.04
		(0.01)	(0.27)	(0.09)	(0.09)	(0.12)
	GARCH-M	0.60	0.06	0.13	0.86		0.46
		(0.17)	(0.03)	(0.02)	(0.02)		(0.03)
Germany	GARCH	2.18	0.40	0.35	0.53
		(0.06)	(0.11)	(0.07)	(0.06)
	GARCH-GJR	2.11	0.44	0.41	0.50	−0.17
		(0.06)	(0.12)	(0.07)	(0.07)	(0.09)
	GARCH-M	0.64	0.11	0.12	0.84		0.57
		(0.31)	(0.04)	(0.04)	(0.04)		(0.15)
Italy	GARCH	4.38	1.27	0.56	0.35
		(0.12)	(0.44)	(0.11)	(0.12)
	GARCH-GJR	4.49	1.37	0.59	0.33	−0.05
		(0.06)	(0.39)	(0.11)	(0.11)	(0.12)
	GARCH-M	1.80	0.04	0.16	0.83		0.22
		(0.20)	(0.02)	(0.04)	(0.04)		(0.04)
Japan	GARCH	1.87	0.70	0.21	0.73
		(0.05)	(0.33)	(0.05)	(0.05)
	GARCH-GJR	1.80	1.52	0.29	0.64	−0.22
		(0.06)	(0.52)	(0.07)	(0.07)	(0.10)
	GARCH-M	−0.36	0.19	0.07	0.91		0.20
		(0.23)	(0.08)	(0.01)	(0.02)		(0.03)
United Kingdom	GARCH	3.61	0.59	0.39	0.57
		(0.08)	(0.25)	(0.07)	(0.07)
	GARCH-GJR	3.98	1.34	0.48	0.49	−0.23
		(0.07)	(0.42)	(0.07)	(0.07)	(0.10)
	GARCH-M	1.75	0.04	0.17	0.82		0.18
		(0.11)	(0.02)	(0.02)	(0.02)		(0.01)
United States	GARCH	2.97	1.10	0.64	0.24
		(0.05)	(0.30)	(0.10)	(0.09)
	GARCH-GJR	2.98	1.30	0.66	0.25	−0.43
		(0.05)	(0.35)	(0.10)	(0.10)	(0.13)
	GARCH-M	2.36	0.72	0.49	0.42		0.06
		(0.18)	(0.24)	(0.10)	(0.10)		(0.02)

Source: Author.

Table 4. Bayesian estimation for the stochastic volatility models: Estimated posterior means (posterior standard error in parentheses).

Countries	Models	α	αh	ρh	${\sigma }_h^2$	ρ	λ
Canada	SV	2.01	1.52	0.97	0.09
		(0.16)	(0.88)	(0.02)	(0.04)
	SV-L	2.01	1.20	0.92	0.08	0.67
		(0.14)	(0.30)	(0.02)	(0.03)	(0.16)
	SV-M	−0.12	1.12	0.97	0.03		0.80
		(0.86)	(0.62)	(0.02)	(0.01)		(0.28)
France	SV	1.97	1.34	0.98	0.09
		(0.12)	(1.11)	(0.01)	(0.03)
	SV-L	1.97	1.11	0.97	0.10	0.15
		(0.12)	(0.86)	(0.02)	(0.04)	(0.13)
	SV-M	−1.84	0.14	0.97	0.02		4.38
		(0.65)	(0.53)	(0.02)	(0.00)		(1.13)
Germany	SV	1.82	0.65	0.95	0.15
		(0.10)	(0.64)	(0.03)	(0.06)
	SV-L	1.81	0.42	0.89	0.13	0.44
		(0.10)	(0.29)	(0.03)	(0.06)	(0.15)
	SV-M	−2.34	−0.63	0.93	0.02		9.22
		(0.57)	(0.29)	(0.03)	(0.00)		(2.62)
Italy	SV	2.42	1.57	0.99	0.11
		(0.11)	(1.23)	(0.01)	(0.03)
	SV-L	2.43	1.15	0.97	0.12	0.17
		(0.11)	(0.95)	(0.02)	(0.04)	(0.13)
	SV-M	−2.92	−0.15	0.97	0.02		8.26
		(0.66)	(0.49)	(0.02)	(0.00)		(2.26)
Japan	SV	0.65	1.94	0.98	0.07
		(0.20)	(1.08)	(0.01)	(0.03)
	SV-L	0.61	1.77	0.96	0.09	0.17
		(0.20)	(0.87)	(0.03)	(0.05)	(0.21)
	SV-M	−1.41	1.26	0.97	0.05		0.94
		(0.34)	(0.62)	(0.02)	(0.02)		(0.19)
United Kingdom	SV	2.27	1.17	0.99	0.12
		(0.12)	(1.25)	(0.01)	(0.04)
	SV-L	2.29	0.95	0.97	0.13	0.14
		(0.12)	(1.01)	(0.02)	(0.04)	(0.14)
	SV-M	0.66	0.64	0.97	0.06		1.46
		(0.31)	(0.72)	(0.02)	(0.01)		(0.34)
United States	SV	2.75	1.21	0.96	0.15
		(0.13)	(0.71)	(0.02)	(0.06)
	SV-L	2.72	1.03	0.92	0.15	0.37
		(0.14)	(0.39)	(0.03)	(0.06)	(0.12)
	SV-M	0.77	0.66	0.98	0.04		0.84
		(0.34)	(0.71)	(0.02)	(0.01)		(0.17)

Source: Author.

Table 5. Log predictive score of two classes of volatility models for both the expanding samples and rolling samples (Canada).

	GARCH	SV	GARCH-GJR	SV-L	GARCH-M	SV-M
Expanding	−84.15	−75.16	−82.29	−73.38	−70.96	−68.16
Rolling	−86.15	−75.47	−84.38	−73.27	−70.94	−69.89

Source: Author.

Table C.1. The posterior estimates of the leverage effect and volatility feedback for other countries. The numbers in parentheses are numerical standard errors.

Country	GARCH-GJR (θ)	GARCH-M (λ)	SV-L(ρ)	SV-M(λ)
Canada	−0.39	0.09	0.67	0.80
	(0.11)	(0.02)	(0.16)	(0.28)
France	−0.04	0.46	0.15	4.38
	(0.12)	(0.03)	(0.13)	(1.13)
Germany	−0.17	0.57	0.44	9.22
	(0.09)	(0.15)	(0.15)	(2.62)
Italy	−0.05	0.22	0.17	8.26
	(0.12)	(0.04)	(0.13)	(2.26)
Japan	−0.22	0.20	0.17	0.94
	(0.10)	(0.03)	(0.21)	(0.19)
United Kingdom	−0.23	0.18	0.14	1.46
	(0.10)	(0.01)	(0.14)	(0.34)
United States	−0.43	0.06	0.37	0.84
	(0.13)	(0.02)	(0.12)	(0.17)
Australia	−0.24	0.32	0.37	0.74
	(0.08)	(0.06)	(0.15)	(0.19)
Austria	−0.13	0.61	0.21	1.90
	(0.07)	(0.08)	(0.15)	(0.63)
Belgium	−0.27	0.25	0.43	2.36
	(0.12)	(0.08)	(0.13)	(0.63)
Denmark	0.00	0.34	−0.05	0.92
	(0.05)	(0.05)	(0.13)	(0.17)
Finland	−0.17	0.33	0.52	3.66
	(0.17)	(0.03)	(0.26)	(1.02)
Luxembourg	−0.17	0.23	0.33	3.40
	(0.08)	(0.05)	(0.15)	(0.95)
Netherlands	−0.01	0.37	0.12	1.34
	(0.09)	(0.07)	(0.14)	(0.38)
New Zealand	−0.66	0.09	0.48	1.61
	(0.13)	(0.01)	(0.16)	(0.55)
Norway	−0.39	0.64	0.24	0.61
	(0.12)	(0.02)	(0.15)	(0.16)
Sweden	−0.25	0.17	0.66	1.26
	(0.09)	(0.02)	(0.18)	(0.24)
Switzerland	−0.25	0.25	0.34	4.74
	(0.15)	(0.05)	(0.11)	(1.40)

Notes: The period spans from 1961Q1 to 2018Q4 for all countries except for Denmark (1967Q1 to 2018Q4).

Source: Author.

Table D.1. Log predictive score of two classes of volatility models for all 18 countries (Expanding samples).

Type	GARCH	SV	GARCH-GJR	SV-L	GARCH-M	SV-M
Canada	−84.15	−75.16	−82.29	−73.38	−70.96	−68.16
France	−93.59	−75.08	−93.96	−74.93	−74.01	−63.83
Germany	−65.20	−59.74	−64.95	−59.13	−56.88	−50.68
Italy	−109.48	−82.66	−109.55	−83.77	−97.38	−71.87
Japan	−97.23	−86.42	−97.12	−86.05	−90.10	−85.63
United Kingdom	−89.77	−64.22	−89.77	−64.93	−65.93	−65.61
United States	−85.22	−84.54	−84.10	−83.08	−82.10	−80.06
Australia	−90.56	−77.24	−89.52	−77.01	−77.89	−76.82
Austria	−71.43	−66.05	−71.17	−65.91	−59.88	−58.74
Belgium	−81.36	−80.85	−81.33	−80.34	−79.98	−75.99
Denmark	−93.60	−73.83	−94.69	−73.69	−72.68	−71.95
Finland	−98.52	−80.44	−99.15	−79.32	−71.16	−70.33
Luxembourg	−79.66	−76.65	−78.95	−76.99	−75.51	−67.04
Netherlands	−83.03	−76.39	−83.35	−76.19	−73.84	−73.72
New Zealand	−108.95	−90.56	−107.45	−90.96	−90.35	−83.45
Norway	−95.93	−79.49	−93.69	−78.52	−80.02	−77.58
Sweden	−101.04	−85.86	−99.10	−82.08	−78.68	−71.19
Switzerland	−88.08	−80.33	−87.20	−82.45	−84.05	−70.52

Notes: The evaluation period is from 2009Q1 to 2018Q4.

Source: Author.

Table D.2. Log predictive score of two classes of volatility models for all 18 countries (Rolling samples).

Type	GARCH	SV	GARCH-GJR	SV-L	GARCH-M	SV-M
Canada	−86.15	−75.47	−84.38	−73.27	−70.94	−69.89
France	−92.86	−73.30	−93.74	−73.46	−71.45	−63.44
Germany	−63.83	−57.64	−63.54	−57.73	−57.48	−49.64
Italy	−109.75	−81.90	−110.05	−82.77	−93.22	−75.42
Japan	−96.48	−86.20	−94.96	−85.68	−89.97	−85.61
United Kingdom	−90.59	−63.38	−90.24	−64.15	−65.62	−63.73
United States	−87.21	−86.35	−87.11	−86.09	−82.56	−80.69
Australia	−93.07	−77.98	−92.67	−77.73	−75.93	−73.95
Austria	−69.75	−64.83	−69.58	−64.58	−60.11	−56.85
Belgium	−81.55	−80.42	−81.44	−79.92	−77.50	−75.31
Denmark	−88.95	−74.76	−89.21	−72.99	−70.65	−69.15
Finland	−96.81	−77.79	−97.04	−76.31	−75.11	−69.01
Luxembourg	−79.85	−76.44	−79.28	−76.68	−73.58	−67.08
Netherlands	−82.36	−75.89	−81.56	−75.77	−72.90	−73.21
New Zealand	−110.51	−89.74	−108.90	−90.41	−88.62	−84.02
Norway	−94.08	−78.50	−92.14	−77.52	−81.89	−76.92
Sweden	−100.61	−82.27	−98.15	−80.01	−76.91	−71.73
Switzerland	−84.55	−76.37	−82.83	−78.70	−81.51	−68.79

Notes: The evaluation period is from 2009Q1 to 2018Q4.

Source: Author.