  While functional regression models have received increasing attention
recently, most existing approaches assume both a linear relationship and a
scalar response variable. We suggest a new method, "Functional Response
Additive Model Estimation" (FRAME), which extends the usual linear regression
model to situations involving both functional predictors, $X_j(t)$, scalar
predictors, $Z_k$, and functional responses, $Y(s)$. Our approach uses a
penalized least squares optimization criterion to automatically perform
variable selection in situations involving multiple functional and scalar
predictors. In addition, our method uses an efficient coordinate descent
algorithm to fit general nonlinear additive relationships between the
predictors and response. We develop our model for novel forecasting challenges
in the entertainment industry. In particular, we set out to model the decay
rate of demand for Hollywood movies using the predictive power of online
virtual stock markets (VSMs). VSMs are online communities that, in a
market-like fashion, gather the crowds' prediction about demand for a
particular product. Our fully functional model captures the pattern of
pre-release VSM trading prices and provides superior predictive accuracy of a
movie's post-release demand in comparison to traditional methods. In addition,
we propose graphical tools which give a glimpse into the causal relationship
between market behavior and box office revenue patterns, and hence provide
valuable insight to movie decision makers.
