How to estimate a ARCH/GARCH model since we don't know last period conditional variance? I'm wondering how it's possible to forecast a conditional variance using a GARCH model since we don't know last period conditional variance.
If I understand correctly, the conditional variance is the variance of the random variable that represent the last period residual of another model. In other terms, we have a model, we have its residual, and we want to forecast that residual variance which mean the variance of all the possible outcome that could have taken the residual, knowing the residual conditional variance last period.
But the problem: we don't have that variance since its the variance of all possible epsilon that could have taken our residual. But in reality, only one residual appeared in the last period. We only have last period residual (required in ARCH model), but we don't have all possible residual that could have happened, knowing last period information set, required to compute the last period conditional variance required for the GARCH model.
Then, how to estimate that last period variance since we don't have the information to do it ?
 A: You have stumbled upon a problem that plagues most of time series analyses. A time series $\{X_t\}$ is an ordered collection of random variables: $X_1,X_2,\dots$. We observe a single observation for a chunk of them: $(x_1,x_2,\dots,x_T)$. Of course, it is impossible to conclude much about the distribution of the random variable $X_t$ from only a single observation $x_t$.
Therefore, we assume some relationship between these distributions, e.g. that all the distributions are the same and independent of each other (the i.i.d. model) or that they are almost the same but dependent with a certain simple structure of how their means (ARMA) or variances (GARCH) are shifting over time (more details here). This makes an observation $x_s$ informative of the distribution of the random variable $X_t$.
In the i.i.d. case, we can treat $(x_1,x_2,\dots,x_T)$ as a sample from $X_1$ and we can obtain an estimate of the mean, the variance and any other features of $X_1$ from the sample. Since $X_2$ and all other random variables have the same distribution, learning about $X_1$ is enough. In the GARCH case, the functional form of the distribution is the same for all $X$s, but the variance is changing as we move from $X_1$ to $X_2$ to ... according to a simple rule. This is enough to be able to learn the parameters of the simple rule, fit the variances of $X_1$ to $X_T$ and predict the variance of $X_{T+1}$. How this is done on the implementation level is not trivial, but the idea behind is based on the logic above.
