Time series analysis: Why am I getting a reciprocal condition number when trying to estimate VAR-Model? /w reproducible example

Question

I'm currently trying to identify an appropriate VARMA(p, q)-representation for a multivariate time series using the MTS::-package in R. The series comprises n = 126 observations representing 12-hour intervals each and k = 14 dimensions. The goal is to identify relations among the series. Specifically, I'm interested in finding out whether the first time series is influenced by the other 13 in a unidirectional relationship.

Prior to the analysis, all series were tested for stationarity using the Augmented Dickey-Fuller- (ADF) and the Kwiatkowski-Phillips-Schmidt-Shin-Test (KPSS). In one case—namely, the suspected output time series of the MTS—ADF- und KPSS-test both suggested that the series was nonstationary, DF = -2.9581, p = 0.1784; KPSS-level = 0.7663, p < .01. However, as the graphical inspection didn't suggest a trend (see Fig. 1), I refrained from differencing the series: For four other series the KPSS-Test suggested nonstationarity, albeit only barely (only just below p = .05). Since the time series plots didn't show any trends and the ADF-test indicated stationarity, I didn't difference these four series, either.

Following Tsay's (2013) procedure, the first step then is selecting a sensible p-order, using

VARorder(da)

which should yield something like

selected order: aic =  5 
selected order: bic =  2 
selected order: hq =  2 

indicating the appropriate p order by consulting various information criteria. However, for my data, the command gives

Error in solve.default(xpx, xpy) : 
  system is computationally singular: reciprocal condition number = 2.92466e-18

suggesting that the matrix containing my data is non-invertibile. Unfortunately, I can't quite figure out why. Possible problems I've ticked off the list:

• m < k: Observations by far outnumber dimensions
• Multicollinearity: I've checked the data by looking for high, significant correlations of individual series—there weren't any. Nevertheless, I tried excluding some series with the highest significant correlations (max r .85, min r = .56 or so). I still received the same error message (obviously with a different condition number).
• Nonstationarity: I've tried excluding the five series mentioned above. I still received the same error message (obviously with a different condition number).

I'd be very grateful if you would share possible explanations and solutions for this error message.

Note: I'm able to check different VAR models and their information criteria "by hand" using, for example

VAR <- VAR(da,
        2, #p order
        output = T)

But I'm not sure why VARorder(da) wouldn't suggest the appropriate VAR order itself.

---

Reproducible Example

library(MTS)
library(tseries)

Part of the dataset:

data <- structure(c(16, 31, 22, 34, 6, 89, 40, 171, 15, 17, 36, 2445, 
1998, 3633, 2413, 4595, 3928, 19897, 21616, 32786, 24915, 33185, 
17234, 30094, 21407, 23071, 11214, 12381, 3504, 26791, 15769, 
16420, 6782, 12417, 7355, 6491, 3617, 3553, 1667, 3613, 2584, 
4256, 2329, 3993, 2033, 2234, 1573, 6873, 3229, 6535, 6087, 3137, 
1641, 1554, 2550, 8849, 3142, 3058, 1285, 2012, 1698, 2634, 1662, 
3193, 2917, 2936, 1484, 1417, 0.875, 7.51612903225806, 0.954545454545455, 
1.61764705882353, 0, 0.134831460674157, 7, 0.461988304093567, 
0.666666666666667, 4.64705882352941, 1.36111111111111, 13.1460122699386, 
8.7967967967968, 7.87723644371043, 4.394944053046, 11.9275299238302, 
10.1517311608961, 9.87390058802835, 7.97141006661732, 4.66802903678399, 
6.98539032711218, 3.49477173421727, 3.26819078565626, 6.65528012228351, 
5.59756154528892, 2.66598760348489, 2.12546816479401, 3.29036426782974, 
3.56449771689498, 5.98637602179836, 3.65660473080094, 3.5247868453106, 
1.4967561191389, 7.78650237577515, 2.5675050985724, 2.49622554305962, 
2.10201824716616, 2.08274697438784, 6.02279544091182, 8.03874896208137, 
3.05882352941176, 9.32894736842105, 4.8196650923143, 5.34510393188079, 
2.49335956714215, 2.26812891674127, 7.04704386522568, 5.14607885930452, 
10.0947661814803, 2.95791889824024, 3.70445211105635, 1.01817022633089, 
3.28153564899452, 2.38095238095238, 17.5701960784314, 5.12532489546841, 
2.08911521323997, 2.69882275997384, 8.70817120622568, 1.1317097415507, 
4.82508833922261, 4.56909643128322, 6.26594464500602, 5.98559348575008, 
10.5279396640384, 4.00340599455041, 1.06469002695418, 1.0317572335921, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.0257668711656442, 0.03003003003003, 
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0.0222365505300353, 0.0203177463933181, 0.0207522869434416, 0.0253700886313811, 
0.0302044114158382, 0.054381401907357, 0.0713482826145553, 0.0612009131968949, 
0.01997317625, 0.0510981161290323, 0.0535885127272727, 0.0252178452941176, 
0.0224598566666667, 0.0366746406741573, 0.03129685, 0.0314023360233918, 
0.0331178286666667, 0.0204349776470588, 0.0457561375, 0.0965838858568507, 
0.0903712287287287, 0.0623022909441233, 0.0528139293410692, 0.0554109867247008, 
0.0436485353869654, 0.0460532286274313, 0.0348924079848261, 0.0353313444763009, 
0.0310355385912101, 0.0307351182763297, 0.0292634714517814, 0.0731096677078487, 
0.0454055047414397, 0.0384545797754757, 0.0389100532370251, 0.0528891220418383, 
0.0338050350171233, 0.0533953797917211, 0.0647036425898916, 0.0532260353227771, 
0.0568478851371277, 0.0296690730450189, 0.0285812043507818, 0.030679913572639, 
0.0334435134089024, 0.0384176397410639, 0.0355680524295141, 0.0277129941876557, 
0.0273872154411765, 0.0841495991541353, 0.0663000777157578, 0.0435628795391936, 
0.0429536104771274, 0.0463072918531782, 0.0305447542911634, 0.021211754837771, 
0.021843973428306, 0.0222774719204285, 0.0177299032364055, 0.0213700710870258, 
0.0283763050578915, 0.0321798532818533, 0.0345026179607843, 0.0326594259238332, 
0.0430079570337365, 0.0410183927403532, 0.0343477150194553, 0.028561967693837, 
0.0349459587161366, 0.0315691636294609, 0.0787937060168472, 0.0552217647979956, 
0.101648102811107, 0.0634827074250681, 0.0437294484501348, 0.0412089564573042, 
0.020856236875, 0.0562083170967742, 0.0308131295454545, 0.0396829629411765, 
0.100044611666667, 0.0456313075280899, 0.04957125, 0.0409887112865497, 
0.0194665993333333, 0.326012851764706, 0.0764910805555556, 0.0588664084867076, 
0.0661107557557558, 0.0637056179466006, 0.0426115840033154, 0.0408393569096844, 
0.039273578385947, 0.0537249937176459, 0.0547690406180607, 0.0606379183187946, 
0.0514832675095324, 0.0479212357993069, 0.0421727962167808, 0.0411807343656543, 
0.0398579445975615, 0.0358253226994929, 0.039641051007669, 0.0322878620466844, 
0.0396163507420091, 0.0345283438468142, 0.0352380429323356, 0.034724115225335, 
0.0309002053966382, 0.0284429056938069, 0.0274001185588035, 0.0347561315667848, 
0.0324393124136024, 0.0407634359696032, 0.0359587619076185, 0.0309502446720177, 
0.0330951520897833, 0.0297507953477444, 0.0458972592099614, 0.0486794437766091, 
0.045119072110182, 0.0354557688898836, 0.0276929989828353, 0.0236847299578059, 
0.0239514215856302, 0.0323983291507269, 0.0246527813372762, 0.0316129636595473, 
0.0329225525289458, 0.0324244047619048, 0.0365879505882353, 0.0652355384789242, 
0.0561048074474857, 0.0688581608894702, 0.0702260007782101, 0.0470446928429423, 
0.0310852744994111, 0.0419289434320425, 0.0312496483754513, 0.040507020357031, 
0.0374511567706548, 0.0364719567438692, 0.0343822626684636, 0.0322565266760762
), .Dim = c(68L, 14L), .Dimnames = list(NULL, NULL))

Test for stationarity using:

adf.test(data[,1])
kpss.test(data[,1])

Identify appropriate VAR(p)-order:

VARorder(data)

Error in solve.default(xpx, xpy) : 
  system is computationally singular: reciprocal condition number = 1.106e-26

Answer 1

You're underestimating the number of parameters required. In a $\text{VAR}(p)$ with $k$ series and no mean, there are $pk^2$ regression coefficients, $pk$ per variable. The function MTS::VARorder tests up to $p=13$, which for $k=14$ gives 2548 parameters, 182 per variable. This is already more than your 126 observations.

You need to reduce the largest $p$ that you'll consider (parameter maxp). When comparing AIC, it has to be done on the same data for all models, which is why you will lose not the first $p$ data points at each $p$, but the largest $p$ for all of them (MTS::VARorder does this automatically). So if that's 13, you effectively only have 113 observations. Take that into consideration when choosing an appropriate largest $p$ to check.