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### Introduction to CADF and Johansen statistical tests for Cointegration

Updated on 2013-06-04 by Guest

Cointegration in econometric time series modeling is defined as that property where two or more time series share a common stochastic drift. To put in more technical terms, if there are two non stationary time series namely X and Y which observe random walks in their movements, then the two series are said to be cointegrated if any linear combination of this series becomes stationary. Cointegration thus moves on to describe a long run equilibrium relationship between the pairs of time series data.

There are many areas in finance where the theory of cointegration could be expected to hold, say for example:

●      Spot and futures price for a given commodity or asset
●      Ratio of relative prices and an exchange rate
●      Equity prices and dividends

In all the three above mentioned examples, market forces arising from no arbitrage assumptions state that there should be an equilibrium relationship between the series concerned. To illustrate, spot and futures may be cointegrated since they are obviously prices for the same asset at different time periods and hence they will get affected in the same way for a given piece of information. The long run relationship between the spot and futures prices of an asset or a share would thus be given by the cost of carry.

Though there are number of econometric applications for cointegration between two time series data, in quantitative finance cointegration forms the fundamental concern for pairs trading strategy adopted by investors. This particular concept is widely used by investors to determine their position of buy or sell between two cointegrated stocks.

For example, if you consider two stocks namely Google Inc. and Yahoo! Inc. which are observed to be cointegrated. The cointegrating relationship between the two stocks might be given by the equation X – 3Y = Z, where X refers to the share price of Google Inc. and Y refers to the share price of Yahoo! Inc.
In this example, Z would refer to a stationary series of zero mean. Thus through the equation, you could understand that the share price of Google Inc. is priced three times higher than the price of Yahoo! Inc. Hence in an arbitrage situation, when the share price of Google Inc. is observed to be more than three times the share price of Yahoo! Inc. then you could be sure that either the share price of Google Inc. would decrease or the share price of Yahoo! Inc. would increase to match and hence result in a no arbitrage assumption for the time series. Thus for an investor with a portfolio of these shares, he could either sell the shares of Google Inc. or buy the shares of Yahoo! Inc. to profit from this situation and the cointegration between the pairs.

Cointegration between pairs of time series data are commonly analyzed using various econometric analysis tools and the popular among them are

2.       Johansen Cointegration tests

The ADF tests for Cointegration were developed by Engle and Granger in 1987 and is thereby also known as the EG test. This test involves running the static regression to test the following equation

Yt = θ’xt + et

In the above mentioned equation, xt is considered to be one or higher dimensional. The pre requisite which ought to be verified before testing this regression equation is that the two time series x and y, both are integrated to the order I(1).
The asymptotic distribution of θ is not standard, but as suggested by Engle and Granger, the value of θ would be estimated using the ordinary least squares regression and the unit roots test would be conducted with the following equation

êt = yt - θ’xt

Since the unit root tests test the null hypothesis of no unit root, the cointegration ADF test would also test the null hypothesis of no cointegration between the time series pairs. In this test, the limiting distribution as given by the Augmented Dickey Fuller test is used for the dimensions of the regressor and hence this EG test is commonly called as the Cointegration ADF tests or CADF tests. Engle and Granger had compared the efficiency of different tests for cointegration and had finally recommended this Cointegration ADF test to be valid. Despite many other tests for unit roots in residuals from potentially cointegration relations between time series, the CADF tests had proven to stand pretty well among the researchers and investors.

2.       Johansen Cointegration Test

Engle--Granger did not permit the testing of hypotheses on the cointegrating relationships themselves, but the Johansen setup does permit the testing of hypotheses about the equilibrium relationships between the variables.

Johansen cointegration test allows a researcher to test a hypothesis about one or more coefficients in the cointegrating relationship by viewing the hypothesis as a restriction on the _ matrix. If there exist r cointegrating vectors, only these linear combinations or linear transformations of them, or combinations of the cointegrating vectors, will be stationary. In fact, the matrix of cointegrating vectors β can be multiplied by any non-singular conformable matrix to obtain a new set of cointegrating vectors.

A set of required long-run coefficient values or relationships between the coefficients does not necessarily imply that the cointegrating vectors have to be restricted. This is because any combination of cointegrating vectors is also a cointegrating vector. So it may be possible to combine the cointegrating vectors thus far obtained to provide a new one or, in general, a new set, having the required properties.

The simpler and fewer are the required properties, the more likely that this recombination process (called renormalization) will automatically yield cointegrating vectors with the required properties.

However, as the restrictions become more numerous or involve more of the coefficients of the vectors, it will eventually become impossible to satisfy all of them by renormalization. After this point, all other linear combinations of the variables will be non-stationary. If the restriction does not affect the model much, i.e. if the restriction is not binding, then the eigenvectors should not change much following imposition of the restriction. Thus the Johansen cointegration test proves to be highly effective to test the hypothesis of existence of equilibrium relationships between the two variables.