Cointegration Search: Variance Minimization and Alternative Methods

Summary

The note addresses why variance minimization can help find cointegrating combinations among nonstationary series. It explains that when a cointegrating relationship exists, the coefficients can be estimated super-consistently, giving more flexibility in how the vector is estimated than ordinary regression settings might suggest.

It describes principal component analysis on series in levels: leading components represent nonstationary directions, while remaining components may identify cointegrating vectors. The answer also mentions regression in levels when one relation is expected, provided residual stationarity is checked, and identifies Johansen’s method as another approach. It offers no empirical comparison or detailed assumptions, so these are method suggestions rather than evidence that one estimator is universally best.

Key ideas

• Cointegrating vectors can be estimated by several methods when a stationary relationship among nonstationary series exists.
• Regression in levels can estimate a candidate relation, but its residuals must be checked for stationarity.
• Cointegration coefficients can be super-consistently estimated under the stated setting.
• PCA on level data may separate nonstationary directions from cointegrating directions.
• Johansen’s method is another commonly used way to estimate cointegrating relationships.

Tags

# Minimizing variance when searching for Cointegration


# Minimizing variance when searching for Cointegration












This paper by Meucci explains that in order to find a combination leading to cointegration of several series $X$, you have to find the vector $w$ which minimise the quantity $\textrm{Var}(w'X)$. I do not understand why we want to minimise variance. Because stationarity needs constant variance and not the smallest possible variance.

## Answer by NBF (score 0, accepted)

https://quant.stackexchange.com/a/41107

There are many ways for finding cointegrating vectors. If we know that there is one cointegration relation, we can just run a regression in levels. (but we have to know our residuals are stationary).

But if the timeseries are cointegrated, then the betas are also super-consistent, which means that $T*(\hat{\beta}-\beta)\rightarrow 0$ as $T\rightarrow\infty$ whereas it is usually only with power $T^{1/2}$ for ordinary OLS. We have lots of leeway in how to estimate our cointegrating vector then.

One method, just to throw you for a loop, is to use a PCA in levels (yes, in levels). The first few principal components will be the non-stationary directions. The remaining ones, will be the cointegrating vectors. This is just one of various methods.

I tend to use Johansen's method, but sometimes simple OLS is just easier.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.