Different results from Trace statistics and Maximum statistic with Johansen cointegration test - vecrank

[Marco Giansoldati]

Dear Members,

I am looking at the rank of cointegration through the vecrank command, checking the relationship between a number of nine variable.

This is the command I am using and the result I got:

. vecrank mgsv iad gdpv dd gdpvx cgv cpv itv xgsv if ifscode==946, max

                       Johansen tests for cointegration                        
Trend: constant                                         Number of obs =      78
Sample:  1995q3 - 2014q4                                         Lags =       2
-------------------------------------------------------------------------------
                                                         5%
maximum                                      trace    critical
  rank    parms       LL       eigenvalue  statistic    value
    0      90      2271.8509           .    280.5480   192.89
    1      107     2304.2582     0.56437    215.7334   156.00
    2      122     2331.1288     0.49792    161.9923   124.24
    3      135     2355.7148     0.46763    112.8202    94.15
    4      146     2374.3496     0.37986     75.5507    68.52
    5      155     2389.0698     0.31439     46.1103*   47.21
    6      162     2400.8646     0.26098     22.5206    29.68
    7      167     2408.2309     0.17211      7.7880    15.41
    8      170     2412.0891     0.09419      0.0717     3.76
    9      171     2412.1249     0.00092
-------------------------------------------------------------------------------
                                                         5%
maximum                                       max     critical
  rank    parms       LL       eigenvalue  statistic    value
    0      90      2271.8509           .     64.8146    57.12
    1      107     2304.2582     0.56437     53.7410    51.42
    2      122     2331.1288     0.49792     49.1721    45.28
    3      135     2355.7148     0.46763     37.2695    39.37
    4      146     2374.3496     0.37986     29.4404    33.46
    5      155     2389.0698     0.31439     23.5897    27.07
    6      162     2400.8646     0.26098     14.7326    20.97
    7      167     2408.2309     0.17211      7.7163    14.07
    8      170     2412.0891     0.09419      0.0717     3.76
    9      171     2412.1249     0.00092
-------------------------------------------------------------------------------

The results of the Trace statistic seems to suggest that there are five cointegrated equations, whereas the Max-Eigen statistic seems to point to the presence of three cointegrated equations.

I went through the theory behind the Johansen test and look for examples when the two statistics indicate different results, but I did not find any evidence.

I would be very grateful if someone could help me in understanding the meaning of such a different outcome.

Many many thanks.

[sakinahman]

I got this information from one forum:
The trace test and the max-eigenvalue co-integration rank tests are different because they test different hypotheses.
If we have p=5 variables the Johansen test involves 5 eigenvalues l₁, l₂, l₃, l₄ and L₅ in decreasing order. There are, for example 3 unit roots if l₃=l₄=l₅=0. In the trace test the alternative is that all 5 eigenvalues are zero. Note that the hypotheses that there are only two or one zero eigenvalues is implied if the 3 hypothesis is accepted. Thus the 3 zero eigenvalue hypothesis implies that there are at least 3 unit values and at most two co-integrating relationships.
In similar circumstances the corresponding maximum eigenvalue test would have an alternative hypothesis that there are 4 zero eigen values. I would be tempted to use this test in circumstances where theory was definite about 1 co-integrating relationship and I wished to test for a second. I do not recall such a scenario in my research.
In other circumstances the trace test is to be preferred. Possible references are Burke and Hunter (2005), Modelling non-stationary economic time series, Palgrave or Enders (2014), Applied Econometric time series, 4th edition, Wiley. Your software makes this look easy but a proper understanding requires a lot of effort.
By John C Frain

[sakinahman]

I have the same problem, in fact exact results as yours, so i decided to use trace test.
Check this:
https://www.researchgate.net/post/In...e_test_differs