# Test for Cointegration Using the Johansen Test

This example shows how to assess whether a multivariate time series has multiple cointegrating relations using the Johansen test.

Load Data_Canada into the MATLAB® Workspace. The data set contains the term structure of Canadian interest rates [142]. Extract the short-term, medium-term, and long-term interest rate series.

Y = Data(:,3:end); % Interest rate data

To illustrate the input and output structure of jcitest when conducting multiple tests, test for the cointegration rank using the default H1 model and two different lag structures.

[h,pValue,stat,cValue] = jcitest(Y,'Model','H1','Lags',1:2);
************************
Results Summary (Test 1)

Data: Y
Effective sample size: 39
Model: H1
Lags: 1
Statistic: trace
Significance level: 0.05

r  h  stat      cValue   pValue   eigVal
----------------------------------------
0  1  35.3442   29.7976  0.0104   0.3979
1  1  15.5568   15.4948  0.0490   0.2757
2  0  2.9796    3.8415   0.0843   0.0736

************************
Results Summary (Test 2)

Data: Y
Effective sample size: 38
Model: H1
Lags: 2
Statistic: trace
Significance level: 0.05

r  h  stat      cValue   pValue   eigVal
----------------------------------------
0  0  25.8188   29.7976  0.1346   0.2839
1  0  13.1267   15.4948  0.1109   0.2377
2  0  2.8108    3.8415   0.0937   0.0713

The default "trace" test assesses null hypotheses $H\left(r\right)$ of cointegration rank less than or equal to r against the alternative $H\left(n\right)$, where n is the dimension of the data. The summaries show that the first test rejects a cointegration rank of 0 (no cointegration) and just barely rejects a cointegration rank of 1, but fails to reject a cointegration rank of 2. The inference is that the data exhibit 1 or 2 cointegrating relationships. With an additional lag in the model, the second test fails to reject any of the cointegration ranks, providing little by way of inference. It is important to determine a reasonable lag length for the VEC model (as well as the general form of the model) before testing for cointegration.

Because the Johansen method, by its nature, tests multiple rank specifications for each specification of the remaining model parameters, jcitest returns the results in the form of tabular arrays, and indexes by null rank and test number.

Display the test results, h.

h
h=2×7 table
r0       r1       r2      Model     Lags      Test       Alpha
_____    _____    _____    ______    ____    _________    _____

t1    true     true     false    {'H1'}     1      {'trace'}    0.05
t2    false    false    false    {'H1'}     2      {'trace'}    0.05

Column headers indicate tests r0, r1, and r2, respectively, of $H\left(0\right)$, $H\left(1\right)$, and $H\left(2\right)$ against $H\left(3\right)$. Row headers t1 and t2 indicate the two separate tests (two separate lag structures) specified by the input parameters.

Access the result for the second test at null rank $r=0$ using tabular array indexing.

h20 = h.r0(2)
h20 = logical
0