Reduced-Rank Envelope Vector Autoregressive Model

Figures & data

Fig. 1 Average estimation errors of moderate-dimensional VAR models with different combinations of $(d,u,p,q)$ against sample size. Refer to Table 1 for details.

Table 1 Total number of parameters (NOP) in each VAR model correspond to Figure 1.

	Large envelop	Nearly full rank	Typical Scenario	p > 1
$(d,u,p,q)$	$(2,6,1,7)$	$(5,6,1,7)$	$(3,4,1,7)$	$(3,4,2,7)$
OLSVAR	77	77	77	126
RRVAR	52	73	61	82
EVAR	70	70	56	84
REVAR	50	68	52	73

Fig. 2 Average estimation errors of higher-dimensional VAR models with different combinations of $(d,u,p,q)$ against sample size. Refer to Table 2 for details.

Table 2 Total number of parameters (NOP) in each VAR model correspond to Figure 2.

	Medium VAR	Medium VAR with $p\mathrm{}>\mathrm{}1$	Large VAR	Large VAR with $p\mathrm{}>\mathrm{}1$
$(d,u,p,q)$	$(5,10,1,20)$	$(5,10,2,20)$	$(10,20,1,40)$	$(10,20,2,40)$
OLSVAR	610	1010	2420	4020
RRVAR	385	485	1520	1920
EVAR	410	610	1620	2420
REVAR	335	435	1320	1720

Fig. 3 Minimum (left panels) and maximum (right panels) asymptotic standard error ratios of coefficient estimates for each VAR model with respect to the proposed REVAR model.

Fig. 4 Average estimation errors for $(d,u,p,q)=(3,4,1,7)$ with Normal, Uniform, t-Student, and Chi-square ${\mathit{\epsilon }}_t$ distributions, plotted against sample size. Refer to Table 1 for details.

Fig. 5 Average estimation errors of moderate-dimensional VAR models with different $(d,u,p,q)$ for stochastic volatility martingale difference sequence (SV-MDS) errors against sample size.

Table 3 Percentage selection of the true dimensions (d and u) and lag order (p).

	$(d,u,p,q)=(3,5,1,7)$			$(d,u,p,q)=(2,4,2,8)$
	p Selection	d Selection	u Selection	p Selection	d Selection	u Selection
T	BIC	Chi-squared test	BIC	BIC	Chi-squared test	BIC
240	100%	93%	88%	100%	83%	54%
450	100%	95%	92%	100%	94%	90%
700	100%	97%	99%	100%	94%	99%
950	100%	98%	100%	100%	93%	98%
1200	100%	98%	100%	100%	94%	99%

Table 4 Pseudo-real-time forecasting performance with bootstrap for the NIPA dataset (1959Q1-2019Q4) using an expanding window scheme, evaluated from 2005Q1 to 2019Q4.

				RMSFEh
$(\widehat{d},\widehat{u},\widehat{p},q)$	Model(M)	NOP	$r_{\text{avg}.}$	h = 1	h = 2	h = 3	h = 4
	OLSVAR	100	1.2912	0.059709	0.059433	0.059647	0.059671
(3, 4, 1, 8)	EVAR	68	1.0854	0.059133	0.05888	0.059074	0.05909
	RRVAR	75	1.0029	0.059278	0.058971	0.059047	0.059086
	REVAR	63	_	0.058950	0.058666	0.058850	0.058788

Table 5 Pseudo-real-time forecasting performance with bootstrap for the Price dataset (1959Q1-2019Q4) using an expanding window scheme, evaluated from 2005Q1 to 2019Q4.

				RMSFEh
$(\widehat{d},\widehat{u},\widehat{p},q)$	Model(M)	NOP	$r_{\text{avg}.}$	h = 1	h = 2	h = 3	h = 4
	OLSVAR	610	1.2236	0.13261	0.13114	0.13214	0.13187
(13, 14, 1, 20)	EVAR	490	1.0203	0.13194	0.13065	0.13156	0.13112
	RRVAR	561	1.1201	0.13228	0.13105	0.13182	0.13157
	REVAR	483	_	0.13184	0.13063	0.13150	0.13101

Table 6 Pseudo-real-time forecasting performance with bootstrap for the initial 43 out of 47 macroeconomic variables in Table S6 (1959Q1-2019Q4), evaluated from 2005Q1 to 2019Q4 using an expanding window scheme.

				RMSFEh
$(\widehat{d},\widehat{u},\widehat{p},q)$	Model(M)	NOP	$r_{\text{avg}.}$	h = 1	h = 2	h = 3	h = 4
	OLSVAR	2795	1.4202	0.21493	0.21371	0.21381	0.21396
(18, 19, 1, 43)	EVAR	1763	1.0406	0.20709	0.20664	0.20735	0.20668
	RRVAR	2170	1.0594	0.21003	0.20956	0.20962	0.20925
	REVAR	1738	_	0.20644	0.20615	0.20681	0.20631