When Pairwise Cointegration May Not Extend Across Three Series

# Co-integration constraints of coint(X,Z) given coint(X,Y) and coint(Y,Z)?


# Co-integration constraints of coint(X,Z) given coint(X,Y) and coint(Y,Z)?












The Augmented Dickey-Fuller Test can be used to measure how well ranked certain pairs are against others for co-integration.

So then say we have a known co-integration between `X` and `Y` and between `Y` and `Z`, is there a constraint to the range of co-integration between `X` and `Z`?

For example, if we know `coint(X,Y) = -0.1` and `coint(Y,Z) = -0.3` can we then use these in some formula which would then say with certainty that `-x.x < coint(X,Z) < -x.x`?

This is similar to the correlation constraint of the same scenario given by (Olkin, 1981).

EDIT - As Richard has pointed out, I may have misunderstood the ADF Test. I'll rephrase the question here: If we know $X$ is cointegrated (via some test e.g. Engle Granger) with $Y$ and $Y$ with $Z$, is there some measure of how probable $X$ will be cointegrated with $Z$?

## Answer by aajajim (score 3, accepted)

https://quant.stackexchange.com/a/10022

Regarding you comments, I'm adding an answer here because I will not have enough space to explain my point, so please forgive for this.

Lets start from the beginning, and assume :

(1) $X_t - \beta_1Y_t = \epsilon_t$ ($\epsilon_t$ is stationary)

(2) $Y_t - \beta_2Z_t = \eta_t$ ($\eta_t$ is stationnary)

then (1) + $\beta_1$(2) gives $X_t - \beta_1\beta_2Z_t = \epsilon_t+\beta_1\eta_t = \nu_t$

Even is $\epsilon_t$ and $\eta_t$ are stationary processes, the linear combination is stationary only if these processes are independent (which is a strong assumption), or in a weak assumption, they should be joint weak stationarity.

Indeed, to be stationary, $\nu_t$ should have an autocovariance which is independent of time $t$, while:

$Cov(\nu_{t+h}, \nu_t) = Cov(\epsilon_{t+h}+\beta_1\eta_{t+h}, \epsilon_t+\beta_1\eta_t) = [Cov(\epsilon_{t+h}, \epsilon_{t})+ \beta_1^2Cov(\eta_{t+h}, \eta_t)] + \beta_{1}[Cov(\eta_{t+h}, \epsilon_t)+Cov(\epsilon_{t+h}, \eta_t)]$

In general, $Cov(\eta_{t+h}, \epsilon_t)$ and $Cov(\epsilon_{t+h}, \eta_t)$ are not only functions of $h$ but also of $t$.

If $\epsilon_t$ and $\eta_t$ are independent, then these covariances are 0, the first ones are independent of $t$ which leads to the desired result. But, as I said, in reality you only need that $\epsilon_t$ and $\eta_t$ be Joint Weakly Stationary, which means that $\forall t_1, t_2~~ Cov(\epsilon_{t_1}, \eta_{t_2})=f(|t_2-t_1|)$ (or $h$ if $t_1=t+h$ and $t_2=t$)

Please find in this link a disscusion about this topic:

https://math.stackexchange.com/questions/377333/sum-of-stationary-process

## Answer by Richi Wa (score 1)

https://quant.stackexchange.com/a/10013

I am trying to give a some comments (too much for a real comment) a start: $X$ cointegrated with $Y$ means there is a $\beta_1$ such that $$ X_t - \beta_1 Y_t = u_t, $$ and $u_t$ is stationary. $\beta_1$ can be estimated by $$ \beta_1 = \frac{Cov(X_t,Y_t)}{Var(Y_t)} = Cor(X_t,Y_t) \frac{\sqrt{Var(X_t)}}{\sqrt{Var(Y_t)}}. $$ For $Y$ and $Z$ we have $$ Y_t - \beta_2 Z_t = v_t, $$ with $v_t$ stationary and $\beta_2 = Cor(Z_t,Y_t) \frac{\sqrt{Var(Y_t)}}{\sqrt{Var(Z_t)}}$.

Then the question is whether there is a $\beta_3$ such that $$ X_t - \beta_3 Z_t = e_t $$ with $e_t$ stationary. We would estimate $$ \beta_3 = Cor(X_t,Z_t) \frac{\sqrt{Var(X_t)}}{\sqrt{Var(Z_t)}}. $$ As linear combinations of stationary time series are stationary we get $$ k_t = u_t - v_t = X_t - \beta_1 Y_t - (Y_t - \beta_2 Z_t) = X_t - (1+\beta_1) Y_t + \beta_2 Z_t $$ and we have trivially found a linear combination of the three that is stationary - this is multicointegration. At the moment I don't know whether this can be simplified - maybe someone else does.