# Burn-in period for random walk

We are trying to make simulation experiment involving a common stochastic trend, that is described by the random walk (or $I(1)$ process) $Y_t = Y_{t-1} + \varepsilon_t$, where innovations $\varepsilon_t$ ~ $N(0,1)$. However when could we be sure that the past innovations are more or less reasonably included into the stochastic trend? Is there any good proxy for random-walk's burn-in period?

I've looked for R's burn-in default suggestion for $AR(p)$ part: ceiling(6/log(minroots)), meaning that in case of unit root we get infinity here and roughly tried to take close to unity root 1.0001 (actually it is the same like taking 60000 at once). So any reasonable suggestions or rule of thumbs you do use in practice?

-
 Could you add a bit more context please? e.g. what R function and package are you using? – onestop Jan 4 '11 at 19:50 A link explaining why random-walk needs a burn-in also would help. – mpiktas Jan 4 '11 at 20:12 Well we will use diffinv() directly, but the burn-in default could be found in arima.sim() function. – Dmitrij Celov Jan 4 '11 at 20:39

If I understand correctly, you want to simulate a stochastic trend. Why not initialize $Y_0$ to a random value, say a sample from $N(0,\sigma^2)$ with $\sigma^2$ large? – vqv Jan 5 '11 at 18:22
Suppose $Y_t = \alpha Y_{t-1} + \epsilon_t$ where $\epsilon_t$ are i.i.d. $N(0,1)$ and $|\alpha| \leq 1$. The conditional distribution of $Y_t$ given $Y_0$ is $N(\alpha^t Y_0, \sigma_{\alpha,t}^2)$, where $\sigma_{\alpha,t}^2 \leq t$ depends only on $\alpha$ and $t$. If $\alpha = 1$, note that $Y_t$ given $Y_0$ remains centered at $Y_0$ even as $t \to \infty$. In that sense, you can always infer $Y_0$. On the other hand, if $|\alpha|< 1$ then the conditional mean of $Y_t$ given $Y_0$ tends to 0 as $t\to\infty$ so that asymptotically $Y_t$ is conditionally independent of $Y_0$. – vqv Jan 6 '11 at 15:42
If $|\alpha| = 1$, then $\sigma_{\alpha,t}^2 = t$. If $|\alpha| < 1$, then $\sigma_{\alpha,t}^2 = (1-\alpha^t) / (1-\alpha)$. So although you can infer $Y_0$ from $Y_t$ when $\alpha = 1$ the standard error will be of the order $\sqrt{t}$. I think to better answer your question you need to be less specific and explain what you will use $Y_t$ for. – vqv Jan 6 '11 at 15:49