# Loss functions with one-step-ahead volatility forecasts & volatility proxy

is anyone here familiar with loss functions like MSE? I have basically 1000 simulated return matrices (T x N where T=700 and N = 5 stocks). Now I have to calculate the one step ahead volatility forecasts on the simulated returns using different models for eg multivariate EWMA,DCC GARCH and input these forecasts into the loss function to select a superior model. I have two questions regarding how to proceed from here:

1) should I calculate the one step ahead volatility forecasts based on the last period's return only? Because both the models mentioned above predict a different covariance forecast for each time period based on the previous time period's forecast.

2) When it comes to choosing a volatility proxy to input in the loss function, the paper I am following mentions that they have used the outer product of error terms (Et' * Et). Now my understanding of loss functions is fairly limited but as far as I know I have to compare the forecasts with the real volatility denoted by the proxy so I am assuming I should take the outer product of error terms on the real return data (as opposed to the simulated returns) but since I have only simulated returns for 700 periods while the actual return data I have is for 1000 periods, should I take the time into consideration when taking the outer product because each time period has a different covariance matrix (1000 periods vs 700 periods)?

• With simulated returns, you know the true underlying volatility. Use it. Using some proxy instead would be inefficient. – Richard Hardy May 4 '17 at 13:01
• Can you please elaborate how I can calculate the true underlying volatility? I'm afraid i am totally new to this so need clarification regarding the implementation. I would really appreciate if you could also tell me why I know the true underlying proxy with simulated returns – Hsk May 4 '17 at 13:03
• You know the true volatility, not a proxy. How do you simulate? You must specify the data generating process when you simulate (it is impossible to do without it). The volatility is part of the specification of the data generating process. Since you specify it, you know it. – Richard Hardy May 4 '17 at 14:11
• Oh okay, now I understand what you meant, thanks for clarifying. Yes I have specified the DGP as DCC GARCH and simulated using the sd calculated by univariate GARCH. However, this paper that I am following has used the outer product of error terms as proxy for the loss function and since I am trying to replicate their study, that's why I want to use the proxy they have used. – Hsk May 4 '17 at 15:22
• That will be a very noisy proxy. Does the study you are refering to give a good reason for using a proxy when they know the true value? Sounds like wasting statistical power. – Richard Hardy May 4 '17 at 15:32

2. Act as if the simulated data is the real data. Use the outer product of the estimated errors (if the conditional mean model is "empty", then just the raw returns) $\varepsilon_{t+1} \varepsilon_{t+1}'$ when assessing the forecast of $\sigma_{t+1}^2$.