Overview
Informatlly, two time series, {% X %} and {% Y %}, are said to be cointegrated if they share a common trend. There are various formal defintions which are designed to capture this notion.
Standard Definition
Time Series {% x_t %} and {% y_t %} are cointegrated if there exist a linear combination of the two that yields a stationary process. That means, that there is an {% \alpha %} and a {% \beta %} such that given the following equation, {% \epsilon_t %} is stationary.
{% y_t - \alpha - \beta x_t = \epsilon_t %}
Common Trend Definition
Two time series, {% y_t %} and {% z_t %} are said to be cointegrated if there are series {% n_{yt} %}, {% \epsilon_{yt} %}, {% n_{zt} %}, {% \epsilon_{zt} %} such that
{% y_t = n_{yt} + \epsilon_{yt} %}
{% z_t = n_{zt} + \epsilon_{zt} %}
where {% n_{yt} %} is non-stationary and {% \epsilon_{yt} %} is stationary.
(and likewise for {% n_{zt} %} and {% \epsilon_{zt} %} )
And also
{% n_{yt} = \gamma n_{zt} %}
That is, the non-stationary (trend) component of one series is equal to some constant times the trend component
of the other series. When this is true, a linear combination of the two series will create a stationary series.
(see Vidyamurthy)