Cointegrated Processes

It is known that time series of various economic indicators can be converted to stationary using differentiation. Generally, estimating a regression that includes such a time series, results in incorrect statistical conclusions that there is a strong relationship between non-related variables.

A non-stationary process, the first differences of which are stationary, is called a first-order integrated process and indicated as Ι(1). A stationary process is indicated as Ι(0).

The I(1)-process Y1t and Y2t are called first-order cointegrated CI(1,0), if there is a linear combination of these processes that is a I(0)-process. In other words, Y1t and Y2t are cointegrated if there is such a coefficient λ, that Y1t - λY2t ~ I(0). A linear combination Y1t - λY2t is named a cointegration equation.

The definition of cointegration naturally includes the case of several cointegrated variables with random order of cointegration. The components of n-dimensional vector I(d)-process Yt = (Y1t, …, Ynt) are called cointegrated components with the order d, b, and indicated as Yt ~ CI(d,b).

The concept of cointegration was introduced by Clive Granger. Time series can be tested for cointegration using several methods, for example, the Engle-Granger two-step method in case of two variables. However, the most common procedure is currently Johansen test.