If x and y are cointegrated, and too y and z, does this mean x and z are also cointregrated Cointegration question.
If x and y are cointegrated, and too y and z, does this mean x and z are also cointregrated.
x, y and z are I(1), the linear combination of x,y is I(0), and the linear combination of y,z is also I(0),
So will the linear combination of x and z also be I(0) and hence cointegrated. 
 A: Not necessarily.
We're given that {x_t}, {y_t}, and {z_t} are all I(1). If processes {y_t} and {x_t} are cointegrated then: $y_t=\beta_0+\beta_1x_t+\epsilon_t$, where $y_t$ and $x_t$  are I(1) and $\epsilon_t$ is I(0). If processes {y_t} and {z_t} are cointegrated then: $y_t=\alpha_0+\alpha_1z_t+e_t$, where $y_t$ and $z_t$  are I(1) and $e_t$ is I(0).
Solving this system of 2 equations for $z_t$ in terms of $x_t$ we get that $z_t=\frac{\beta_0-\alpha_0}{\alpha_1}+\frac{\beta_1}{\alpha_1}x_t+\frac{\epsilon_t-e_t}{\alpha_1}$
$z_t$ and $x_t$ are cointegrated if $\frac{\epsilon_t-e_t}{\alpha_1}$ is an I(0) process. Since $\alpha_1$ is a scalar, our conclusion rests on whether $\epsilon_t-e_t$ is I(0), where both $\epsilon_t$ and $e_t$ are I(0). We could rephrase it to "Is a linear combination of 2 I(0) processes I(0)?". The answer to this question appears to be "not necessarily"; it depends on whether the joint distribution of the two processes is the same for all t. See here for more on this sub-question: https://math.stackexchange.com/questions/377333/sum-of-stationary-process
