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if $n = 1$, then the return is the RV as well. :-(. I am using it to compare the simulated conditional variance of GARCH model. In other words, I want to get absolute error between RV and simulated ones and then get a conclusion about which GARCH model is better to describe the volatility.
Thanks @ocstl. So if $n = 10$, it is just a ten days' realised volatility and it is impossible to estimate the daily volatility with only the daily returns. it that true? well, what I need is daily volatility, so I need 5 min or 1 hour returns to estimate the daily volatility?
Thanks @SRKX. I understand it now. So one more question, $\bar{Q}$ can be estimated by the standard residuals, so is this matrix is the parameters in the model, i n other words, I want to estimate the Bayesian Information Criterion (BIC), and I need to know the number of parameters, I wonder whether the three parameters in the matrix needs to be considered. If not considered, then there will be 8 parameter in a bivariate case, otherwise, there will be 11 parameters.
No, @Jean-Paul, I have minus the mean of the time series preliminarily, so they are two models to describe the volatility. So my method is like Kupiec test but just to calculate the violated rate.
I just count the times the return cross the threshold(VaR, alpha = 5%) and then divide the total observation number to get a probability p. If the more p is close to 5%, the more the model accurate. I am not sure this method works.
Yes, I just need to make a comparison. One more thing, if it is OK for you, @Jean-Paul. I have BIC for model A and model B and it shows A is better fit. But in the VaR test, the case is converse, is that reasonable?
@John, random walk is one kind of Metropolis Hasting method as we still need to compare the density of proposed and current parameters, the only difference is that the random walk save your much time to calculate the proposed distribution as the normal distribution is symmetric. Forget to express my thanks to you about MLE. ^^
Rejecting negative also could be used in the random walk, for example, alpha + x, x is sampled from a normal distribution, if alpha + x is negative, reject and re-sample till it is positive, true?
@John, thanks so much. Actually, I can ensure that alpha and beta are positive in the random walk as they are sampled from a normal distribution and I can reject the negative ones. By the way, could you recommend me a method in MLE? I mean an algorithm to find the optimization point, I first tried Newton's method but I hate second partial, especially when it comes to the multivariate GARCH models, it is annoying to me. And I also tried gradient method, which only requires first order, do you have some method that can be useful in fit these model but do not need have second or high order?
