Sample output from xsim_varma_fit_var.r, which uses the R MTS package to simulate 1000 observations of a bivariate VARMA(1,1) process and fit VAR (vector autoregression) models of successively higher orders, is below. The BIC and HQ criteria choose a VAR with 3 lags, fewer than the AIC, which chooses 4.
Simulated process coefficients:
AR Coefficients (phi):
[,1] [,2]
[1,] 0.2 0.3
[2,] -0.6 1.1
MA Coefficients (theta):
[,1] [,2]
[1,] -0.5 0.0
[2,] 0.0 -0.6
Covariance Matrix (sigma):
[,1] [,2]
[1,] 4.0 0.8
[2,] 0.8 1.0
dim(zt): 1000 2
Autocorrelations of series 'zt', by lag
, , Series 1
Series 1 Series 2
1.000 ( 0) 0.528 ( 0)
0.674 ( 1) 0.254 (-1)
0.284 ( 2) 0.043 (-2)
, , Series 2
Series 1 Series 2
0.528 ( 0) 1.000 ( 0)
0.627 ( 1) 0.936 ( 1)
0.601 ( 2) 0.812 ( 2)
lag.ar = 1
Constant term:
Estimates: -0.1044549 -0.03062638
Std.Error: 0.06916922 0.0362292
AR coefficient matrix
AR( 1 )-matrix
[,1] [,2]
[1,] 0.475 0.223
[2,] -0.564 1.114
standard error
[,1] [,2]
[1,] 0.0249 0.01476
[2,] 0.0130 0.00773
Residuals cov-mtx:
[,1] [,2]
[1,] 4.6827073 0.9880773
[2,] 0.9880773 1.2846616
det(SSE) = 5.039397
AIC = 1.625287
BIC = 1.644918
HQ = 1.632748
lag.ar = 2
Constant term:
Estimates: -0.08525757 -0.009309301
Std.Error: 0.06679199 0.03348674
AR coefficient matrix
AR( 1 )-matrix
[,1] [,2]
[1,] 0.655 0.233
[2,] -0.597 1.512
standard error
[,1] [,2]
[1,] 0.0334 0.0637
[2,] 0.0167 0.0319
AR( 2 )-matrix
[,1] [,2]
[1,] -0.226 -0.0342
[2,] 0.231 -0.4516
standard error
[,1] [,2]
[1,] 0.0508 0.0697
[2,] 0.0254 0.0350
Residuals cov-mtx:
[,1] [,2]
[1,] 4.3414079 0.9168607
[2,] 0.9168607 1.0912584
det(SSE) = 3.896964
AIC = 1.376198
BIC = 1.41546
HQ = 1.39112
lag.ar = 3
Constant term:
Estimates: -0.0910338 -0.02056322
Std.Error: 0.06622381 0.0326565
AR coefficient matrix
AR( 1 )-matrix
[,1] [,2]
[1,] 0.699 0.217
[2,] -0.601 1.603
standard error
[,1] [,2]
[1,] 0.0346 0.0691
[2,] 0.0171 0.0341
AR( 2 )-matrix
[,1] [,2]
[1,] -0.363 0.0088
[2,] 0.285 -0.7722
standard error
[,1] [,2]
[1,] 0.0640 0.1193
[2,] 0.0316 0.0588
AR( 3 )-matrix
[,1] [,2]
[1,] 0.154 -0.0262
[2,] -0.105 0.2513
standard error
[,1] [,2]
[1,] 0.0543 0.0756
[2,] 0.0268 0.0373
Residuals cov-mtx:
[,1] [,2]
[1,] 4.2461172 0.8878123
[2,] 0.8878123 1.0325305
det(SSE) = 3.596035
AIC = 1.303832
BIC = 1.362725
HQ = 1.326215
lag.ar = 4
Constant term:
Estimates: -0.08472878 -0.01682396
Std.Error: 0.06614913 0.03262521
AR coefficient matrix
AR( 1 )-matrix
[,1] [,2]
[1,] 0.708 0.238
[2,] -0.604 1.627
standard error
[,1] [,2]
[1,] 0.0350 0.071
[2,] 0.0173 0.035
AR( 2 )-matrix
[,1] [,2]
[1,] -0.377 -0.0431
[2,] 0.305 -0.8450
standard error
[,1] [,2]
[1,] 0.0671 0.1304
[2,] 0.0331 0.0643
AR( 3 )-matrix
[,1] [,2]
[1,] 0.204 0.0533
[2,] -0.140 0.3950
standard error
[,1] [,2]
[1,] 0.0698 0.1311
[2,] 0.0344 0.0646
AR( 4 )-matrix
[,1] [,2]
[1,] -0.0456 -0.0588
[2,] 0.0517 -0.1043
standard error
[,1] [,2]
[1,] 0.0552 0.0775
[2,] 0.0272 0.0382
Residuals cov-mtx:
[,1] [,2]
[1,] 4.2140872 0.8794244
[2,] 0.8794244 1.0250894
det(SSE) = 3.546429
AIC = 1.297941
BIC = 1.376465
HQ = 1.327786
lag.ar = 5
Constant term:
Estimates: -0.08584221 -0.02093731
Std.Error: 0.06632399 0.03263429
AR coefficient matrix
AR( 1 )-matrix
[,1] [,2]
[1,] 0.708 0.236
[2,] -0.603 1.630
standard error
[,1] [,2]
[1,] 0.0352 0.0713
[2,] 0.0173 0.0351
AR( 2 )-matrix
[,1] [,2]
[1,] -0.381 -0.0339
[2,] 0.307 -0.8658
standard error
[,1] [,2]
[1,] 0.0679 0.1332
[2,] 0.0334 0.0656
AR( 3 )-matrix
[,1] [,2]
[1,] 0.211 0.0351
[2,] -0.149 0.4420
standard error
[,1] [,2]
[1,] 0.0735 0.1436
[2,] 0.0362 0.0706
AR( 4 )-matrix
[,1] [,2]
[1,] -0.0612 -0.0205
[2,] 0.0620 -0.1857
standard error
[,1] [,2]
[1,] 0.0714 0.134
[2,] 0.0351 0.066
AR( 5 )-matrix
[,1] [,2]
[1,] 0.0220 -0.0287
[2,] -0.0164 0.0572
standard error
[,1] [,2]
[1,] 0.0555 0.0779
[2,] 0.0273 0.0383
Residuals cov-mtx:
[,1] [,2]
[1,] 4.2150939 0.8790021
[2,] 0.8790021 1.0205030
det(SSE) = 3.528871
AIC = 1.300978
BIC = 1.399133
HQ = 1.338284
Results:
Lag AIC BIC HQ
1 1.625287 1.644918 1.632748
2 1.376198 1.415460 1.391120
3 1.303832 1.362725 1.326215
4 1.297941 1.376465 1.327786
5 1.300978 1.399133 1.338284
Best lag according to AIC: 4
Best lag according to BIC: 3
Best lag according to HQ: 3
The script xsim_var_fit_var.r simulates from a bivariate VAR(1) process with only 30 observations. In this case
only BIC chooses the correct lag order.
Results:
Lag AIC BIC HQ
1 1.2742610 1.461087 1.3340284
2 1.4718569 1.845509 1.5913915
3 1.3075726 1.868052 1.4868746
4 1.1098720 1.857177 1.3489414
5 0.6248741 1.559006 0.9237108
6 0.4598563 1.580814 0.8184604
7 0.5678868 1.875671 0.9862582
8 0.6714662 2.166077 1.1496049
Best lag according to AIC: 6
Best lag according to BIC: 1
Best lag according to HQ: 6