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I have two time-series a & b. The objective is to find out whether two series are cointegrated or not. I am using Johansen Test in R to find this out.

I am using urca package of R.

Here is the summary of test (trace test with constant intercept): ca.jo(cbind(a,b), type="trace", ecdet = "const", K = 2, spec ="longrun")

Summary:

Johansen-Procedure

Test type: trace statistic , without linear trend and constant in cointegration

Eigenvalues (lambda):

[1] 1.729720e-02 4.118294e-03 1.294090e-19

Values of teststatistic and critical values of test:

             test  10pct 5pct  1pct
    r <= 1 | 2.46  7.52  9.24  12.97
    r = 0  | 12.88 17.85 19.96 24.60

Eigenvectors, normalised to first column: (These are the cointegration relations)

            a.l2          b.l2         constant
a.l2        1.000000     1.0000000     1.000000
b.l2       -3.662895     0.6463026     1.725186

constant 1135.666923 -2889.4155208 -7862.128714

Weights W: (This is the loading matrix)

       a.l2         b.l2        constant
a.d    0.002621493 -0.006226421 1.245608e-18
b.d    0.010169925 -0.001446919 2.187151e-18

Now my question how to interpret this result and determine whether a & b are cointegrated or not? What is a loading matrix in a cointegration test? How to interpret the critical values? How to determine whether to keep a constant intercept or zero intercept? Do I need to check individual series is an I(1) series before running johansen test?

There is a similar question which has been asked before here but it didn't answer my question completely.

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1 Answer 1

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Some of your question was already answered on the question you mention. Please read it carefully to understand better. In particular it answers very well how to conclude if there is co-integration or not. Also note that this question is not really relevant here both on level and subject (It is a pure statistical question and can be asked on stats.stackexchange.com). If you need more detail and proofs on that subject you could read Johansen seminal article: Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models. (It is very technical though)

Now Let's take the other one by one.

1/The loading matrix is the matrix generaly reffered to as alpha (Check urca documentation).

2/ The critical values: If the null hypothesis (r=0, r<=1) is verified your test statistic follows a known distribution. Given the cumulative distribution you can find where lie 90%, 95%, 99% of the values. Here under the null your test statistic (the trace) is distributed a chi^2. Therefore if it's value is greater than some of the critical values you can reject the null at this confidence. Obviously in your case you cannot reject anything at any confidence (doesn't mean that you proved the null is verified). I'm not telling you if this means co-integration or not as it is much better that you find that out for yourself.

3/ I'm not so sure about the intercept (in the VECM) but it is critical as it corresponds to a deterministic trend in the VAR representation and changes your test statistics. I suppose you could first fit a model with th intercept and test for it's significance. My belief is that deterministic trend is not very probable with financial time series.

4/ Contrary to the tests (ADF and others) based on Engle and Granger methodology you do not need to test if your series are I(1) previously as this is one of the null in your trace test. Chek which one in the previous question you mentioned.

As a rule I think anyone should try to apply : Don't use a statistical method if you don't understand it.

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  • $\begingroup$ Thank you so much for you reply. I still have a few doubts. In 4th point, are you saying that in johansen test we do not need to test series is I(1) individually? Can you take one hypothesis (r=0 or r<=1) and explain the null hypothesis and the exact criterion when to reject it. And also, in chi square distribution, what should be the degree of freedom I should use. Thanks again! $\endgroup$
    – techpaisa
    May 25, 2012 at 5:51
  • $\begingroup$ @techpaisa you have to successively test, the first one is null: r=0 against r>0. in your case you reject at 99% if your test (12.88) >24.60. If this was true you would then test r=1 against r>1 : 2.46>12.97. if you reject that then you can conclude that r=2. You do not need to know the degree of freedom as the critical values are tabulated $\endgroup$
    – Zarbouzou
    May 25, 2012 at 8:53
  • $\begingroup$ I thought we have to make sure we're not entering any $I(0)$ variables into the Johansen procedure? Introductory Econometrics for Finance begins their Johansen chapter with 'Suppose that a set of $g$ variables are under consideration that are $I(1)$ and ....'. Also wouldn't you just get trivial cointegrating relationships if you enter in $I(0)$ variables? $\endgroup$
    – Jase
    Dec 28, 2012 at 6:27
  • $\begingroup$ I think you might want to elaborate on your point 4, Tsay in his Multivariate TSA book prepended ADF test when demonstrating Johansen Test precisely to ensure the input vectors were nonstationary. ca.jo result doesn't seem to show you any information w.r.t the order of the input vectors? what is the null in trace test showing whether input vecotrs are of integrated process (and its order) ? $\endgroup$
    – stucash
    Dec 26, 2018 at 12:39

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