  A projective moving average $\{X_t, t \in \mathbb{Z}\}$ is a Bernoulli shift
written as a backward martingale transform of the innovation sequence. We
introduce a new class of nonlinear stochastic equations for projective moving
averages, termed projective equations, involving a (nonlinear) kernel $Q$ and a
linear combination of projections of $X_t$ on "intermediate" lagged innovation
subspaces with given coefficients $\alpha_i, \beta_{i,j}$. The class of such
equations include usual moving-average processes and the Volterra series of the
LARCH model. Solvability of projective equations is studied, including a nested
Volterra series representation of the solution $X_t$. We show that under
natural conditions on $Q, \alpha_i, \beta_{i,j}$, this solution exhibits
covariance and distributional long memory, with fractional Brownian motion as
the limit of the corresponding partial sums process.
