B/VAR - Insignificance of non-diagonal AR parameters

[KonstantinV]

I am trying to estimate a Macroeconomic VAR/BVAR for the purpose of forecasting/impulse response simulations.

I have a total of 5 macro variables that are 4 country specific and 1 for the Euro Area :

GDP_BG - GDP ; HICP_BG - Inflation; HP_BG - House prices ; Unemp_BG - Unemployment ; GDP_EA - GDP euro area

I have colelcted quarterly and monthly data for the period 2006 - 2016. After cubic interpolation of the quarterly data all series are in monthly year on year grwoth format.

1. DF tests indicate non-stationarity of the series:

Dickey-Fuller Unit Root Tests
Variable	Type	Rho	Pr < Rho	Tau	Pr < Tau
GDP_BG	Zero Mean	-8.62	0.0402	-1.95	0.0495
	Single Mean	-15.03	0.0348	-2.64	0.0885
	Trend	-17.17	0.108	-2.93	0.1564
GDP_EA	Zero Mean	-42.7	<.0001	-4.44	<.0001
	Single Mean	-48.69	0.0011	-4.75	0.0002
	Trend	-48.02	0.0004	-4.73	0.001
Unemp_BG	Zero Mean	-0.18	0.6399	-0.22	0.6043
	Single Mean	-5.76	0.36	-1.77	0.3947
	Trend	-7.8	0.5888	-2.33	0.4161
HICP_BG	Zero Mean	-1.13	0.4495	-0.48	0.5071
	Single Mean	-3.39	0.6051	-1.18	0.6833
	Trend	-12.61	0.2665	-2.51	0.3215
HP_BG	Zero Mean	-13.83	0.0087	-2.61	0.0093
	Single Mean	-14	0.0458	-2.61	0.0929
	Trend	-14.24	0.1953	-2.74	0.2225

2. Taking this into account I run the VAR(3) - suggested by the minimum information criteria

/* VAR with 1st Difference*/
proc varmax data=DATA_ALL.MACRO_CLEAN_ALL;
   model GDP_BG GDP_EA Unemp_BG HICP_BG HP_BG / p=3 noint dftest dify=(1)  
                 print=(estimates);
run;

3. The DF test now indicates stationarity, however, there is no significant relationship apart from the diagonal elements of the AR parameters, except for higher order lags:

AR Coefficient Estimates
Lag	Variable	GDP_BG	GDP_EA	Unemp_BG	HICP_BG	HP_BG
1	GDP_BG	1.51695	-0.32323	-0.07316	0.0208	0.04657
	GDP_EA	0.04184	1.54188	-0.0317	-0.00889	0.02111
	Unemp_BG	0.0585	-0.13691	1.01772	0.02646	-0.01353
	HICP_BG	-0.09687	-0.17317	0.45338	0.26286	0.05087
	HP_BG	0.40168	-0.6717	-0.16238	0.07366	1.52713
2	GDP_BG	-1.24578	0.57102	0.24424	-0.007	-0.01004
	GDP_EA	-0.01199	-1.08216	0.046	-0.00174	-0.01859
	Unemp_BG	-0.03327	0.1483	-0.32875	-0.02099	0.01321
	HICP_BG	0.02011	1.41164	-0.48313	-0.00346	-0.11958
	HP_BG	-0.27895	2.18608	-0.45604	0.22938	-1.32891
3	GDP_BG	0.4411	-0.33352	-0.25461	-0.04826	0.01708
	GDP_EA	0.02513	0.38162	-0.01667	-0.01701	0.01481
	Unemp_BG	-0.02864	0.10291	-0.01005	-0.03522	0.00014
	HICP_BG	0.31382	-1.4971	-0.1214	0.14482	0.10065
	HP_BG	0.35618	-2.09016	0.63977	0.11498	0.54739
Schematic Representation of Parameter Estimates
Variable/Lag	AR1	AR2	AR3
GDP_BG	+....	-....	+....
GDP_EA	.+...	.-...	.+...
Unemp_BG	..+..	..-..	.....
HICP_BG	...+.	.....	.-...
HP_BG	....+	...+-	.-..+
+ is > 2*std error,  - is < -2*std error,  . is between,  * is N/A

If I proceed with imposing a BVAR prior the  higher order lags are eliminated by the prior and the resulting model has significant parameters at low order diagonal parameters.

I don't understand the reason for these results. How is it possible that stationary macro variables do not have any dependence between each other? Any suggestions for what am I making wrong and what could I try?

Could the monthly frequency be the reason for these results? If so should I test much higher lag orders (p=12) and then restrict the insgignificant parameters?