A functional dependent variable \(y_i(t)\) is approximated by a single
functional covariate \(x_i(s)\) plus an intercept function \(\alpha(t)\),
and the covariate can affect the dependent variable for all
values of its argument. The equation for the model is
$$y_i(t) = \beta_0(t) + \int \beta_1(s,t) x_i(s) ds + e_i(t)$$
for \(i = 1,...,N\). The regression function \(\beta_1(s,t)\) is a
bivariate function. The final term \(e_i(t)\) is a residual, lack of
fit or error term. There is no need for values \(s\) and \(t\) to
be on the same continuum.