I'm coming at this from a finance and engineering background. I don't have the depth of statistical knowledge to infer this one completely myself, I apologize for that. I'll probably need some hand holding through this since the 30 websites about likelihood equations still seem rather Greek to me. HA! can't be the first time someone used that pun on this site!
I'm trying to build some forecasting models for 48 months of financial time series data on more than 50 accounts. For each account I would like to use the previous 48 months to develop a model and then forecast for an additional 12 months into the future. I know, all Models are False, Some are useful. I'm hoping to use AIC to help me infer the model that is the most likely to be useful. Now, this gets more exciting and makes AIC more important when I start trying to add volume data and seasonality as well as alternate models such as logistic growth... but I digress. Basics first. Date is X, Value is Y.
For now I just need to get the basic idea behind a Log Likelihood comparison between a 2nd and 3rd order polynomial. Now, i understand the idea of using a Chi-squared test to compare the differences between the models based on the number of parameters in each model (2 and 3 respectively). But i DON'T understand what the Likelihood equation is for a polynomial! How do I determine the Likelihood equation for a polynomial? For some reason the vertical bar included in all of the examples is throwing me through a loop.
AIC = 2k-2*ln(L)
where k is the number of parameters
and ln(L) is the likelihood equation
How do I determine the likelihood equation for a second and third order polynomial?