Inquiry on Johansen method (jcitest function)

Dear all,

I've got a quick question with regard to the code segment in

http://www.mathworks.co.kr/kr/help/econ/identify-cointegration.html#btdp660

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load Data_Canada Y = Data(:,3:end); % Interest rate data [~,~,~,~,reg] = egcitest(Y,'test','t2'); c0 = reg.coeff(1); b = reg.coeff(2:3); beta = [1; -b];

[~,~,~,~,mles] = jcitest(Y,'model','H1*'); BJ2 = mles.r2.paramVals.B; c0J2 = mles.r2.paramVals.c0;

% Normalize the 2nd cointegrating relation with respect to % the 1st variable, to make it comparable to Engle-Granger: BJ2n = BJ2(:,2)/BJ2(1,2); c0J2n = c0J2(2)/BJ2(1,2);

% Plot the normalized Johansen cointegrating relation together % with the original Engle-Granger cointegrating relation:

COrd = get(gca,'ColorOrder');

plot(dates,Y*beta-c0,'LineWidth',2,'Color',COrd(4,:)) hold on plot(dates,Y*BJ2n+c0J2n,'--','LineWidth',2,'Color',COrd(5,:)) legend('Engle-Granger OLS','Johansen MLE','Location','NW') title('{\bf Cointegrating Relation}') axis tight grid on hold off

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********************** Results Summary (Test 1)

Data: Y Effective sample size: 40 Model: H1* Lags: 0 Statistic: trace Significance level: 0.05

r h stat cValue pValue eigVal ---------------------------------------- 0 1 38.8360 35.1929 0.0194 0.4159 1 0 17.3256 20.2619 0.1211 0.2881 2 0 3.7325 9.1644 0.5229 0.0891

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According to the test result, it implies that there is only one cointegrating relation. Yet there exist two columns in BJ2 which means that only one column of two is a candidate for the single cointegrating relation.

At this point, the code simply assumes that it is the second column for the relation, which justifies the following part.

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% Normalize the 2nd cointegrating relation with respect to % the 1st variable, to make it comparable to Engle-Granger: BJ2n = BJ2(:,2)/BJ2(1,2); c0J2n = c0J2(2)/BJ2(1,2);

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Now my question is how do you know the cointegrating relation is the second column, not the first one?

Please let me know.

Thanks in advance!

Answer 1

[Hang Qian]

I think the number of cointegrations inferred from the data is suggestive rather than conclusive. If we have some theory that backs up the belief that there are 2 cointegrations among these three variables, then trust that belief. My understanding is that the demo illustrates that the EG cointegration matches one of the two cointegrations (or some linear combinations of the two) estimated by Johansen’s FIML, though I am not sure if we can always find the same on all other datasets.

[Hyungseok Hahm]

Hi Hang,

Thanks for the answer. My background is not exactly in the econometrics and thus I am learning by myself. Your answer is very helpful. Please allow me follow-up questions.

(1) The above example clearly suggests that there is only one cointegrating relation in that r0 = 1 AND r1 = 0. Yet BJ2 has 2 potential cointegrating relations which are equivalent to 2 column of the matrix(BJ2n). Then somehow the example assumes that the single cointegrating relation turns out to be the second column, given that the segment of the code, BJ2n = BJ2(:,2)/BJ2(1,2). Any idea how the author comes to the conclusion? Is there any way to find out which column of the matrix corresponds to the single cointegrating relation?

(2) The result summary in the example suggests that there is a single cointegrating relation, but suppose that there were two cointegrating relations(r0 = 1 AND r1 = 1 AND r2 = 0). If so, does it imply that each column of the matrix(BJ2) (and linear combination of the columns, of course) corresponds to each of the two cointegrating relations? The example in the following link (<http://www.mathworks.co.kr/kr/help/econ/jcitest.html>) sort of backs up my theory.

Please enlighten me. Thanks in advance!

[Hang Qian]

For the first question: no, I do not expect the estimated single integrating relation fits any of the columns, since it could be a linear combination of the multiple cointegrations, if the sample size were infinity. Actually, if we invert the dependent and explanatory variables in the EG regression, we could obtain different estimation on the cointegration coefficients, for a given dataset. I tend to take the estimated cointegration results as a reference, not a golden rule.

For the second question: yes, I think you are right. Each column of the matrix refers to a cointegration relation produced by Johansen’s full information maximum likelihood. The JCITEST is both a hypothesis testing tool and an estimation tool.