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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 ?

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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.

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  • $\begingroup$ Thank you for your answer, this is well explained. I'm just not able to fully understand precisely how we can "fit" that model : for example, I do understand the fitting process of an AR(1) model, X_t = X_{t-1} + epsilon_t, because we already have X_t, we already have X_{t+1}, the "fitting" process will reduce as much as possible the sum of the epsilon^2 (for OLS), or increase the sum of log prob. of the actual value VS the predicted value (for MLE), because we do have these values at all times. $\endgroup$
    – Jerem Lachkar
    Commented Dec 23, 2022 at 20:58
  • $\begingroup$ Now considering a GARCH specification of that same AR(1), in order to fit the model, the first question I say to myself is : what are the actual values and the predicted values at all times ? Responses : the actual values are the conditional variances at all times (that we don't have, we have only epsilon at all times), and the predicted values are built from a process that needs 2 things : (1) epsilon_{t-1} at all time (for the ARCH, we have it) and (2) sigma_{t-1} (for the GARCH, we can have it with the process, but we still need an initial variance to compute the first term) $\endgroup$
    – Jerem Lachkar
    Commented Dec 23, 2022 at 20:59
  • $\begingroup$ So in the end in my mind, to sum up, I have 2 missing informations to "fit" the model using MLE: (1) the actual conditional variances that the model needs to approach, and (2) the initial conditional variance required to build the estimated variances from the process. In the end, the process of "fitting" a model is (whatever approach we're using) to find a vector of parameters so that the estimated values are the closer possible (in prob, or distance) to the actual values. My confusion here comes from the fact that we don't have directly these 2 informations $\endgroup$
    – Jerem Lachkar
    Commented Dec 23, 2022 at 21:01
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    $\begingroup$ @JeremLachkar, in a typical implementation of the MLE estimator of a GARCH model, the initial conditional variance is guessed. The initial values of the parameters of the conditional variance equation are also guessed. We start from there and then iteratively move away from the starting values using the gradient to determine the direction we should go in. $\endgroup$
    – Richard Hardy
    Commented Dec 24, 2022 at 7:30

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