# Variance of a recursed substition function

So, I have this function: $$y_t=v_t+\rho v_{t-1}+\rho^2 v_{t-2}+\dots+\rho^{t-1}v_1+\rho^ty_0$$

And I want to find the variance (and after that the covariance, but I should be able to do that..). I know that the variance is: $Var[y_t]=\frac{var[v]}{1-\rho^2}=\frac{\sigma^2_v}{1-\rho^2}$

But somewhere I mess everything up : (

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You used the F-word! Not that it bothers me much, but others here are more easily offended... (suggestion: replace it with mess). Also, try to fix your latex (you probably want to write v_{t-1} to get $v_{t-1}$). –  t.b. Apr 4 '11 at 10:45
This is to second Theo's comments and to signal that the question has no simple answer unless one specifies the dependence structure of $(v_t)$. Is $(v_t)$ an independent sequence? Also you say you want to find the variance but you already know what the variance is? Maybe you want to understand how one can find the variance. –  Did Apr 4 '11 at 10:57
@user9079: I tried to fix your displayed equation, but I don't know if the result is the intended result. To get a compound exponent/index, enclose the entire exponent/index in curly brackets. That is, to get $v_{t-2}$, type v_{t-2}, and not v_t_-_2. –  Arturo Magidin Apr 4 '11 at 15:23
Oh, I am sorry! I am used to another version of LaTeX where you can use _ everytime. And about the swearing, I am even more sorry. I come from a culture where swearing is very common. To @Didier: You are spot on in every comment. This is what happens with too much coffee, too little IQ and too hard classes. Sorry. To @Arturo Magidin thank you for correcting it! You guys are nice! I guess this is why I usually do social science, if you don't know the answer, you just write something that sounds smart.. –  Eirik Apr 6 '11 at 11:16
@Eirik What a piece of advertisement for social science... :-) What is your specific field? –  Did Apr 6 '11 at 22:19

You probably assume that the sequence $(v_t)$ is i.i.d. and that $(v_t)$ and $y_0$ are independent. Write $s_t$ for the variance of $y_t$ and $u$ for the variance of $v_t$. The variance of an independent sum is the sum of the variances, hence $$s_t=u+\rho^2u+\cdots+\rho^{2t-2}u+\rho^{2t}s_0=\frac{1-\rho^{2t}}{1-\rho^2}u+\rho^{2t}s_0,$$ where the second formula assumes that $|\rho|<1$. To get your formula, one can either assume that $t\to+\infty$, or that $(y_t)$ is stationary. In the first case, $\rho^{2t}\to0$ hence $s_t\to u/(1-\rho^2)$. In the second case, $s_t=s_0$ hence plugging this value of $s_0$ in the formula above, one gets $s_t=u/(1-\rho^2)$ for every $t$.