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.

• This is related to the fact that, if X1 and X2 are cointegrated, regressing X1 on X2 leads to a consistent estimate of the cointegration vector. And regression is looking for least squares, remember? – stans Aug 4 '18 at 12:09
• Empirically, regressing X1 on X2 and regressing X2 on X1 does not give the same result. Is it because I have a finite number of observations ? With an infinite number I would get the same result ? Ok I understand the link with regression. The eigenvector associated with the smallest eigenvalue is the vector which leads to the smallest variance and so to the minimum of a least square problem, right ? Thank you for your help. – JeanGuillaume Aug 6 '18 at 10:34
• Yes, you get two different answers for X1 ~ X2 and X2~ X1. This is related to one of the properties of least squares estimation. However, both vectors are consistent estimates of the cointegrated relationship. To identify the better one, you can run ADF scoring on the residuals. – stans Aug 6 '18 at 13:36

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.

• When you say "in levels" do you mean that for example with stocks I should use the price and not the returns ? Or do yo mean to use Multilevel Regression ? My original question was : why PCA leads to cointegration ? Or why does this method work ? – JeanGuillaume Aug 6 '18 at 11:59
• Cointegration is about using I(1) variables together in a linear combination to find an I(0) combination. We almost never think of returns as being I(1). Prices are. Returns are usually thought of as being (locally) covariance stationary. So cointegration of two equities in pairs should mean you could regress one on the other + a constant, and the residuals should be stationary (e.g., mean reverting). – NBF Aug 6 '18 at 13:53
• Now, if you have a bunch of cointegrated assets, then they are basically a linear combination of some I(1) and I(0) variables. There is an alternative formulation of coint called Stochastic Common Trends, i.e., if you take Y our I(1) variables, Y(t) = L F(t) + eps(t) where F are the stochastic common trends, and L^perp would be the cointrating vectors. If you do PCA on Y, you search out the highest variance directions, right?. Well, I(1) variables have higher variance than do I(0). (well they will most definitely if you increase the horizon of course). This is the intution behind it. – NBF Aug 6 '18 at 13:55
• Ok thanks for the intuition. It is what I was looking for. About the higher variance for I(1) series, do you have research paper about it or is it just an empirical fact ? And do you have references for the convergence (in case of cointegration) of the estimated Beta toward the "real" Beta ? – JeanGuillaume Aug 6 '18 at 14:02
• For convergence rates, just search google for super consistent estimator or similar and you'll find it. For higher variance, this is absolutely intuitive, since the var(B_t-B_0) ~ O(t) for a Brownian motion, while var(X_t - X_0) ~ O(1) for a stationary variable. That's basically the definition. For references, I like the (old) book by Maddala and Kim. No proofs, but big survey of the lit as of that point in time. – NBF Aug 6 '18 at 14:25