Confused with basic time series terminology I am a bit confused with the basic time series terminology:
Consider the following words:


*

*fitted values

*forecasted values

*in-sample forecasts

*out-of-sample forecasts

*in-sample fit


I am using GARCH models in my thesis. So the equations have a lagged character.
E.g. GARCH(1,1): $\sigma_t^2=\alpha_0+\alpha_1\epsilon^2_{t-1}+\beta_1\sigma_{t-1}^2$
My fitted values are the $\hat{\sigma}^2_{T+1|T}$ with using in-sample data, i.e.
I have estimates of the alpha and beta (estimation using all in-sample data) and I plug in the in-sample data. Due to the lagged character of the equation this could also be called in-sample forecasts, right? This would be  $\hat{\sigma}^2_{T+1|T}$, $\hat{\sigma}^2_{T+2|T}$, $\hat{\sigma}^2_{T+3|T}$, $\hat{\sigma}^2_{T+4|T}$....
Forecasted values in general means forecasting the volatility equation at the end of the time horizon, so this is out-of-sample, right? This would be out-of-sample forecasts?
I am not sure about the meaning "in-sample fit". I mean, fit is always in-sample right? Or otherwise this would imply that there is an out-of-sample fit, which makes no sense? Does the word combination in-sample fit exists??
Is it right, that I can use fitted and forecasted values or in-sample forecasts and out-of sample forecasts? Because I think that fitted values is the same as in-sample forecasts and forecasted values is equal to out-of-sample forecasts
 A: The term fit usually refers to fitting a model by estimating parameters.  The 'fitted' model parameters are found using in sample data, and applied to generate fitted values across all samples. Forecasted (usually used in time series terminology) values refer to the unknown out of sample response variables estimated from the fitted model. Most prediction literature will use fitted or predicted values for both in sample and out of sample data as both are dependent variable estimates based on the fitted model.

Forecasted values in general means forecasting the volatility equation
  at the end of the time horizon, so this is out-of-sample, right? This
  would be out-of-sample forecasts? 

Beyond the fitted time horizon, yes.  Out of sample, in this case, yes.  But if you were using something like cross-validation to fit a model, various folds prior to the forecasted time horizon could use both in sample and out of sample descriptions.  In machine learning, the term test data, is reserved for virgin untouched data.

I am not sure about the meaning "in-sample fit". I mean, fit is always
  in-sample right? Or otherwise this would imply that there is an
  out-of-sample fit, which makes no sense? Does the word combination
  in-sample fit exists??

No, see above; the fit refers to model selection, estimated parameters, and estimated response values used across all samples. While the fitting is done in sample, the final fitted models are used across all samples. Maybe it is clearer to use in sample (model) fitted values, and out of sample (model) forecasted values to delineate between fitted regions for time series.

Is it right, that I can use fitted and forecasted values or in-sample
  forecasts and out-of sample forecasts? Because I think that fitted
  values is the same as in-sample forecasts and forecasted values is
  equal to out-of-sample forecasts

I would stay with using fitted to refer to models, estimated parameters, and estimated dependent values across all samples.  Reserve forecasting for time series future, unknown response variables.  I think some of the confusion comes from the usage of in sample and out of sample terminology, which comes from various machine learning and statistical learning fields. Most of the time series texts have not caught up with that terminology that I know of.
A: The word fitted may be more suitable when you deal with prediction (prediction pertains to in sample observations: predicted values are calculated for the observations in the sample used to estimate the regression) and forecast may be more suitable when you deal with out-of-sample. See the similar post here.
