L36 Cointegration & Further

Spurious Regression¶

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

Non-spurious regression¶

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 $I(1)$¹, to remove the stochastic trend, we take the first difference of both and then we note than the relationship is not significant ($v_{t}$ and $u_{t}$ are independent).

Cointegrated¶

\[ y_{t} - \dfrac{x_{t}}{0.6} = u_{t}^*, \quad u_{t}^* \sim WN(0, 1+0.6^{-2}) \]

There exists a $\beta = (1,-(0.6)^{-1})$ such that,

$$
\beta'\begin{pmatrix}
y_{t} \
x_{t}
\end{pmatrix} \sim I(0)
$$
Then $x_{t}$ 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 ($I(0)$)

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

Existence of long-run equilibrium
Common stochastic trend exists
Separate the short- and long- run relationships
help improve long-run forecast accuracy (how?)
improve estimation efficiency by implying restrictions on the model (what?)

Multivariate Situations¶

Elements of $k$-dimensional vector $Y$, are cointegrated of order $(d,c)$
if all $Y$ elements are $I(d)$
There exists a non-trivial linear combination $z$
which is $I(d-c)$

\[ \beta'y_{t} = z_{t} \sim I(d-c) \]

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

Applications¶

Pairs Trading in finance: individually assets are not stationary. Linear combination is! There will be some long-run equilibrium between two assets. So 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

If a process contains an $I(1)$ process, it, itself will become $I(1)$. ↩