I'm a beginner student of time series as applied in economics. When we don't have a model for the dependent variable, how do we define the error term ? If we had a model, the difference can be seen from the predicted term vs real value ..
I would say that you can calculate your error for the mean when you have no model at all. The term baseline is used for this. You may be familiar with the notion of training and testing data sets. Where we use the rule of thumb of 80 20 and take the training to be the 80% of the data and testing to be the remaining 20%. The idea is that you train a model on the training dataset and test it on the testing to check for errors. You do this by using RMSE. Root mean square of error.
P represents the predicted value using your model. Your question is as I understand what if we have no model. In that case, we use the baseline model, which is simply comparing with the average. Here is an example.
(if you are using R, this might be useful since it's an R code. y is simply the dependent variable that we want to predict.See there is no model at all. We calculate it based on the average of the training dataset. If we had the model, we would use predicted values of the model instead of the average. Hope this was useful.
The error term whether the model be AR or MA or ARIMA or a Transfer Function simply reflects your ignorance/lack of understanding as to one or more omitted but potentially important predictor variable. It could also mean that you have either under-modelled or over-modelled and thus have either omitted structure or injected structure into the error term