I'm pretty new to Statistics (my technical background is slightly different) so apologies if this question seems dull. I got thrown head first into an applied math based project. What is the difference between innovations, and residuals when discussing GARCH? And how would one find the innovations in order to fit GARCH parameters? My understanding is that we calculate variance(t)=a0+a1(variance(t-1)^2)+b1(returns(t-1)^2). But if we calculate variance throughout a time series of data, what's stopping our variance from going towards infinity (for example) since it seems like it has no basis in reality given that the equation is dependent only on our previous calculation (especially if returns(t)=volatility(t)x error(t)). Once we have a time series of variances, we can use student-t distribution with by fitting the innovations (whatever that even means), yet the innovations are said to be 'the difference of predicted values and observed values', yet our variance only relies on previously calculated variances. Any help is appreciated. This has been driving me crazy.
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2$\begingroup$ You might benefit from reading a book chapter on GARCH (from some time series textbook) where the model is introduced in a pedagogical manner. $\endgroup$– Richard HardyCommented Oct 8, 2020 at 20:21
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$\begingroup$ Thank you for your suggestion! $\endgroup$– AUser240Commented Oct 8, 2020 at 21:16
1 Answer
+1 on reading a book chapter on GARCH, to understand model construction from scratch. Your question is basically answered by looking at any algorithm that fits the model to real-world data. To keep it short, there is indeed nothing that stops the variance to explode, hence in practice the model parameters are restricted to a particular range ensuring stationarity. Then using some distribution for the shocks, a likelihood is formulated, basically we try to figure out which parameter values are the most likely given the data observed. This is indeed non-trivial due to the interaction between "residual" and "innovation" described by your question, since a recursive relation is embedded (changing the parameter will change both the residual per period together with the thing we identify as "innovation").
