Difference between "estimated" and "fitted"? Currently I am using a r package to fit an ARMA-GARCH process. Afterwards, I want to use the fitted values to calculate the Value at Risk. So these values are not the forecasts, but the "in-sample-fit". Now, I am wondering about the word-choice, i.e. the difference between the word "fitted" and "estimated"? What word should I use? Is it the same?
E.g. Tsay page 136 says in figure 3.8 

"Time series plot of estimated volatilty ($\sigma_t$) for monthly
  excess returns ...."

Whereas in the R manual it says: 

"fitted values"

 A: I think the term "fitted" is mostly used in regression frameworks where you fit a line through data. For example: $\hat{y}=\beta_{0}+\beta_{1}x$. The value of the fitted line at the data point $x_{i}$ is then called the fitted value or the predicted value at this point ($\hat{y_{i}}$). The term estimate is in generally used whenever you estimate a population parameter - that might be a slope, mean, standard deviation etc. - from a sample. For example: we say we estimate the population mean $\mu$ by the sample mean $\bar{x}$, so $\hat{\mu}=\bar{x}$. Parameter estimates are usually denoted by a hat operator: $\hat{\sigma}$, $\hat{y_{i}}$, $\hat{\mu}$ etc. So the predicted value in a regression ($\hat{y_{i}}$) can also be seen as an estimate at the point $y_{i}$.
A: Often you want to fit a model, not values. The estimate of parameters of this model are then estimated using an estimator, that is a specific estimation technique. For example in Tsay, the standard deviation (volatility) is a parameter that is estimated. It sounds clear to me. In the R manual, "fitted values" sounds very unclear. Are they speaking about evaluations from an already fitted model?
