Cointegrating Processes¶
Tendency of 2 variables (that are theoretically at equilibrium) to be related to each other
2 processes that are integrated of order 1, but \(\exists\) linear combination of the 2 variables that is stationary.
If there is divergence, it will only be temporary, as there is bound to be error correction
The coefficient associated with the 2 variables will be non-zero
Usually happens with highly connected variables
If there are \(n\) cointegrating variables, then there can be
- \([1, n-1]\) independent cointegrating relationships (not lesser or greater than this range)
- \([1, n]\) error correction relationships
eg:
- Demand and Supply for a commodity
- US interest rate and UAE interest rate
- US is leading market
- UAE is following market
- Dubai and Sharjah rent
- GCC stock markets
Consider \(x, z\) which are both \(I(1)\) processes; \(x_t\) and \(z_t\) are cointegrated processes \(\iff u_t\) is stationary process,
\[ \begin{aligned} z_t &= \alpha_1 x + u_t & \text{(Long-Term Specification)} \\ \implies u_t &= z_t - \alpha_1 x_t & \text{(Short-Term Specification)} \\ z_t - z_{t-1} &= \textcolor{hotpink}{-}\alpha_D(z_{t-1} - \alpha_1 x_{t-1}) + v_t \\ \Delta z_t &= \textcolor{hotpink}{-}\alpha_D(u_{t-1}) + v_t \\ & \text{if } x \text{ also has correcting tendancy,} \\ \implies \Delta x_t &= \textcolor{orange}{+} \alpha_G(u_{t-1}) + w_t \end{aligned} \]
- \(\alpha_D\)
- Speed of adjustment parameter, or error correction coefficient
- \(\alpha_D \in (0, 1)\)
- \(u_t=\) Disequilibrium error/Cointegration residual
Parts¶
- Attractor/Leader
- Attracted/Follower
Correlation vs Co-integration¶
Co-integration \(\ \not \!\!\!\!\! \iff\) Correlation
| Correlation | Co-Integration | |
|---|---|---|
| Co-movement Duration | short-term | long-term |