# Innovations/Residuals/Errors in statistics or GARCH

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.

• You might benefit from reading a book chapter on GARCH (from some time series textbook) where the model is introduced in a pedagogical manner. Commented Oct 8, 2020 at 20:21
• Thank you for your suggestion! Commented Oct 8, 2020 at 21:16