Cointegration Proofs
The proofs below are what I thought about a long time ago when I was a master student.
Proposition 1.1. Let $x_{t}$ ~ I(1), $y_{t}$ ~ I(1) and $z_{t}$ ~ I(1). Suppose $x_{t}$ and $y_{t}$ are cointegrated and that $x_{t}$ and $z_{t}$ are co-integrated. Then, $y_{t}$ and $z_{t}$ are co-integrated.
(Proof)
Let $x_{t}$ + $ky_{t}$ ~ I(0) and $x_{t}$ + $k΄z_{t}$ ~ I(0). Then, it is claimed that $ky_{t}$ − $k΄z _{t}$~ I(0).
Suppose $ky_{t}$ − $k΄z_{t}$ ~ I(1). Then, ($x_{t}$ + $k΄z_{t}$) + ($ky_{t}$ − $k΄z_{t}$) = $x_{t}$ + $ky_{t}$ ~ I(0). But the left-hand side of the equation is the sum of I(0) and I(1), which is I(1). Thus we have a contradiction.
Corollary 1.2. Let $x_{t}$ ~ I(1), $y_{t}$ ~ I(1) and $z_{t}$ ~ I(1). Suppose $x_{t}$ and $y_{t}$ are cointegrated and that $y_{t}$ and $z_{t}$ are co-integrated. Then, $x_{t}$ and $z_{t}$ are co-integrated.
(Proof)
Use proposition 1.1. $y_{t}$ and $x_{t}$ are co-integrated , and $y_{t}$ and $z_{t}$ are co-integrated. Then, $x_{t}$ and $z_{t}$ are co-integrated.
Proposition 1.3. Let $x_{t}$ ~ I(1), $y_{t}$ ~ I(1) and $z_{t}$ ~ I(1). Suppose $x_{t}$ and $y_{t}$ are cointegrated and that $x_{t}$ and $z_{t}$ are co-integrated. Then, $x_{t}$ , $y_{t}$, and $z_{t}$ are co-integrated.
(Proof)
Let $x_{t}$ + $αy_{t}$ ~ I(0) and $x_{t}$ + $βz_{t}$ ~ I(0). From the previous proposition, $αy_{t}$ − $βz_{t}$ ~ I(0).Now it is claimed that $x_{t}$ + $2αy_{t}$ − $βz_{t}$ ~ I(0). That is, $x_{t}$, $y_{t}$, and $z_{t}$ are co-integrated. Suppose not. Then,
$x_{t}$ + $2αy_{t}$ −$βz_{t}$ ~ I(1).
− ($αy_{t}$ − $βz_{t}$) + ($x_{t}$ + $2αy_{t}$ −$βz_{t}$) = $x_{t}$ + $αy_{t}$~ I(0).
However, the left-hand side of the equation
is the sum of I(0) and I(1), which is I(1). Thus we have a contradiction.