Are there empirical models that predict variance? Empirical models, like regressions, generally assume the existence of errors in the data, and effectively ignore them during prediction. This results in an optimised mean projection, but it means that the projection of multiple values will have a lower variance than the data being predicted.
I guess you could just model the errors as well, and add those to the regression predictions, which would increase the variance, but it would also reduce the accuracy of the mean performance. I guess this is inevitable to some degree.
Are there models that are explicitly designed for projection of variance, rather than projection of the mean? Are the models that attempt to do both?
 A: I would say that the general notion of "Empirical models, like regressions, [...] effectively ingore them (the errors) during prediction" is not right. Typically only half of the work is done with a point estimate. It is an integral part of statistics to provide inference and uncertainity quanitfication. If we assume vor example a simple  gaussian regression process 
$$ Y_i=X_i\beta +\epsilon_i $$ with $\epsilon_i\sim i.i.d. N(0,\sigma^2) $, $\beta$ being a paramter vector and $X_i$ being a vector of covariates for subject $i$. 
Then after fitting the paramters on data $(Y_i,X_i,i=1,...,n)$ we obtain estimates $(\hat{\beta},\hat{\sigma}^2)$. If our goal is now the prediction of values $(Y_j, j=n+1,...,N)$ based on covariates $(X_j, j=n+1,...,N)$, we are using only the first moment of our process, i.e.
$$ E[Y_j|X_j]=X_j\hat{\beta}. $$ It follows that prediction itself is by definition free of the variance measure. However, the process of estimation implies that $\hat{\beta}$ itself is a random vector as it is a function of random variables. Hence if we want to consider the variance of the prediction $X_j\hat{\beta}$ we have to correct for the variance of the estimator $V(\hat{\beta})=\hat{\sigma}^2(X^TX)^{-1}$ that is a function of the estimated variance of the errors and the variance within the $X_i$'s for $i=1,...,n$.
Hence the goal is not to provide predictions that have the same variation as the original data (this is impossible as the prediction focusses on the expectation) but to provide predictions together with a prediction error range that mirrors the variation observed in the original data.
When it comes to models that explicitely allow for modelling the variance as well as forcasting it based on previous realizations you can look at so called (G)ARCH models (see here on wikipedia and here a script on volatility/variance forecasting) that are often used to model volatility clusters in finance.
