AR(2) model is causal AR(2) model is:
$$X_t=\phi_1X_{t-1}+\phi_2X_{t-2}+W_t$$
Where $W_t\sim N(o,\sigma^2)$
I want to prove AR(2) model is causal. So, I tried as:
$$X_t-\phi_1X_{t-1}-\phi_2X_{t-2}=W_t$$
$$\Rightarrow (1-\phi_1 B-\phi_1 B^2)X_t=W_t$$
where $B$ is back-shift operator, i.e., $B X_t=X_{t-1}$
$$\Rightarrow X_t=(1-\phi_1 B-\phi_1 B^2)^{-1}W_t$$
Then I don't know how can I proceed?
 A: A famous theorem (Theorem 3.1.1., Brockwell, Davis. Time series: theory and application）states that an ARMA($p$, $q$) process 
$$\phi(B)X_t = \theta(B) W_t$$
is causal if and only if $\phi(z) \neq 0$ for all $z \in \mathbb{C}$ such that $\left|z\right|\leq 1$. 
So in order the AR($2$) process to be causal, the coefficients $\phi_1$ and $\phi_2$ must satisfy
$$1 - \phi_1 z - \phi_2 z^2 \neq 0$$
for all $\left|z\right| \leq 1$. It is not a causal process for all $\phi_1, \phi_2$. For example, $\phi_1 = 2, \phi_2 = 0$.
A: your final equation leads to the MA representation of an AR process.
Pred[X(t)] = cons + a1*W(t-1) + a2*W(t-2) + .... an*W(t-n) reflecting how previous errors "cause" X.
All ARMA models can be presented as pure AR models (weighted average of the past )
or as a pure MA mode ( weighted average of the past errors )
A: You write:
"I want to prove AR(2) model is causal."
Is simply not possible. AR and/or ARMA models are never causal. ARMA models was thinked exactly for describing a process with its own past. These explicitly have merely statistical meaning.
Causality is something the go beyond merely statistical relationship and involve more than one variable. If we are not aware about this, we surely conflate statistical association and causality. At most you can ask about Granger causality (unhappy name) but the univariate nature of ARMA put away this possibility too. 
For these reasons I disagree with previous answers that give you other informations without warning about causal meaning. 
A: AR(2) is causal if :
$$ \phi_1+\phi_2 < 1$$
and
$$ \phi_1 - \phi_2 < 1$$
and 
$$ -1 < \phi_2 < 1$$
In this conditions $ \phi_2(B)=0 $ equation roots are outside of unit circle so it's causal.
