Cointegration & Further¶

Cointegration is a fundamental concept in multivariate time series that addresses the relationship between non-stationary variables. It allows us to distinguish between misleading correlations and genuine long-term equilibrium relationships.

Spurious Regression¶

Consider two independent $I(1)$ processes:
$$x_{t} = x_{t-1} +v_{t},\quad v_{t} \sim WN(0,1)$$
$$y_{t} = y_{t-1} +u_{t},\quad u_{t} \sim WN(0,1)$$

Even though they are generated from independent families, there is a correlation $\implies$ Spurious regression. This means the relationship is just a mathematical artifact of the trends rather than a real connection.

Non-spurious regression¶

Now consider two dependent $I(1)$ processes:
$$x_{t} = 0.6I_{t} + v_{t},\quad v_{t} \sim WN(0,1)$$
$$y_{t} = 0.6I_{t} + u_{t},\quad u_{t} \sim WN(0,1)$$
where $I_{t} =I_{t-1} + w_{t}$ and $w_{t} \sim WN(0,1)$ which is an $I(1)$ process.

Thus, $I_{t}$ is a common stochastic trend in both $\{ x_{t} \}$ and $\{ y_{t} \}$. Since $x_{t}$ and $y_{t}$ are $I(1)$ (because if a process contains an $I(1)$ process, it itself will become $I(1)$), to remove the stochastic trend, we take the first difference of both. After differencing, we note that the relationship is not significant because $v_{t}$ and $u_{t}$ are independent.

Cointegrated¶

In the non-spurious example, we can see:
$$y_{t} - \dfrac{x_{t}}{0.6} = u_{t}^*, \quad u_{t}^* \sim WN(0, 1+0.6^{-2})$$

There exists a vector $\beta = (1,-(0.6)^{-1})$ such that:
$$\beta'\begin{pmatrix} y_{t} \\ x_{t} \end{pmatrix} \sim I(0)$$

When this happens, $x_{t}$ and $y_{t}$ are said to be cointegrated. The series should be transformed so that they can be considered as realizations of weakly stationary processes ($I(0)$).

Definition

When the linear combination ($\beta'(y_{t}, x_{t})'$) of two $I(1)$ processes is weakly stationary $I(0)$, it implies cointegration.

Why is this important?
- Existence of long-run equilibrium: They move together.
- Common stochastic trend exists: One trend drives both.
- Separation of relationships: It helps separate the short- and long-run relationships.
- Improved Forecasts: It helps improve long-run forecast accuracy because the model knows they must stay together.
- Estimation Efficiency: It improves efficiency by implying restrictions on the model parameters.

Multivariate Situations¶

Elements of a $k$-dimensional vector $Y$ are cointegrated of order $(d,c)$ if:
- All $Y$ elements are $I(d)$.
- There exists a non-trivial linear combination $z$:
$$\beta'y_{t} = z_{t} \sim I(d-c)$$

$\beta$ is called the cointegration vector.
There can be multiple non-trivial linear combinations (they need not be unique).
The number of such linearly independent cointegration vectors is called the cointegration rank.

Applications¶

Pairs Trading in finance: Individually assets are not stationary, but their linear combination is! There will be some long-run equilibrium between two assets. One can capture the deviations from the long-run equilibrium in the short-term and take decisions accordingly (by taking long- and short- positions appropriately).
Energy markets: Prices of energy commodities are non-stationary but often move together.