ISSUE

The issue is about cointegration and the interpretation of trace test and eigenvalue test.

JOHANSEN TEST

The Johansen test is a test for cointegration of several I(1) time series data. “Cointegration” is the property of two time series data where both share common stochastic drift. Stochastic drift is the change in average value of the random or stochastic process. The advantage of the Johansen test comes from its ability to handle several time series variable; in contrast, the Engle-Granger test could handle only one cointegration relationship. The Johansen test relies on two types of tests: (i) trace test and (ii) maximum eigenvalue test. Assume that there is a vector autoregression (VAR) of order p:

(1)   yt = mu + A1yt-1 + … + Apyt-p + et

Where y is n x 1 (n by 1) vector that are integrated of order 1 or commonly written as I(1), and et is n x 1 vector of innovations. Equation can be rewritten as:

(2)   ^yt = mu + Pyt-1 + sum(Gi^yt-I) + et

… where P = sum(Ai – I) and Gi = -sum(Aj)

The coefficient P has a reduced rank r < n; thus, there exists n x r matrices alpha (a) and beta (b) with rank r such that P = ab’ and b’y is stationary. Recall that r is the number of relationships. The elements of a is known as the adjustment parameters in the vector error correction model (VECM), b is the vector column of cointegrating vector. The objective is to show that for a given r “what is the likelihood estimator of b defining the combination vector yt-1 that produces r the largest conanical correlations of ^yt with yt-1?” In answer this question the trace test and eigenvalue test are used.

(4)   Jtrace = -T(sum(ln(1 – Li))

(5)   Jmax = -T ln(l - Lr+1)

… where T = sample size, L is the largest canonical correlation. Under the trace test, the null hypothesis of r conintegrating vectors against the alternative hypothesis n cointegrating vectors. Under the eigenvalue test, the null hypothesis of r conintegrating vector against the alternative hypothesis of r + 1 cointegrating vectors.

INTERPRETATION

From the two tests, read the largest canonical correlation of ^yt with yt-1. Both trace test and eigenvalue test are based on pure unit root assumption. When the variables are near-unit-root process, this assumption will not be correct. The interpretation is the interpretation of correlation. However, in this case, the correlation is a correlation for vector variables in time series. The objective of the test is to verify if the two variables---in this case spot price and future price---have a common stochastic drift. The rationale of the test is that any variable that is not I(1) or a pure unit root process is a stationary I(0) process.

REFERENCES:

Johansen, Søren (1991). "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models". Econometrica 59 (6): 1551–1580. JSTOR 2938278.

ISSUE

The issue is about cointegration and the interpretation of trace test and eigenvalue test.

JOHANSEN TEST

The Johansen test is a test for cointegration of several I(1) time series data. “Cointegration” is the property of two time series data where both share common stochastic drift. Stochastic drift is the change in average value of the random or stochastic process. The advantage of the Johansen test comes from its ability to handle several time series variable; in contrast, the Engle-Granger test could handle only one cointegration relationship. The Johansen test relies on two types of tests: (i) trace test and (ii) maximum eigenvalue test. Assume that there is a vector autoregression (VAR) of order p:

(1)   yt = mu + A1yt-1 + … + Apyt-p + et

Where y is n x 1 (n by 1) vector that are integrated of order 1 or commonly written as I(1), and et is n x 1 vector of innovations. Equation can be rewritten as:

(2)   ^yt = mu + Pyt-1 + sum(Gi^yt-I) + et

… where P = sum(Ai – I) and Gi = -sum(Aj)

The coefficient P has a reduced rank r < n; thus, there exists n x r matrices alpha (a) and beta (b) with rank r such that P = ab’ and b’y is stationary. Recall that r is the number of relationships. The elements of a is known as the adjustment parameters in the vector error correction model (VECM), b is the vector column of cointegrating vector. The objective is to show that for a given r “what is the likelihood estimator of b defining the combination vector yt-1 that produces r the largest conanical correlations of ^yt with yt-1?” In answer this question the trace test and eigenvalue test are used.

(4)   Jtrace = -T(sum(ln(1 – Li))

(5)   Jmax = -T ln(l - Lr+1)

… where T = sample size, L is the largest canonical correlation. Under the trace test, the null hypothesis of r conintegrating vectors against the alternative hypothesis n cointegrating vectors. Under the eigenvalue test, the null hypothesis of r conintegrating vector against the alternative hypothesis of r + 1 cointegrating vectors.

INTERPRETATION

From the two tests, read the largest canonical correlation of ^yt with yt-1. Both trace test and eigenvalue test are based on pure unit root assumption. When the variables are near-unit-root process, this assumption will not be correct. The interpretation is the interpretation of correlation. However, in this case, the correlation is a correlation for vector variables in time series. The objective of the test is to verify if the two variables---in this case spot price and future price---have a common stochastic drift. The rationale of the test is that any variable that is not I(1) or a pure unit root process is a stationary I(0) process.

REFERENCES:

Johansen, Søren (1991). "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models". Econometrica 59 (6): 1551–1580. JSTOR 2938278.

Respected Paul Sir,

                        I sincerely acknowledge for having clarified my doubts on Co-Integration.

Heartiest Thanks.

have a look at the Appendix A. Representation of the Johansen maximum likelihood (ML) estimator - I think you will find this useful and complementary to Paul's eplanation. 

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Respected Valeri Sir,

                 Thanks for sending the reference paper. Sincere Thanks.