we can rewrite (5) as Then the only parameter that remains to be estimated is the regression slope β$_{0}$ . Since { φ$_{k}$ } is an orthonormal basis of the X -space and { ψ$_{l}$ } is an orthonormal basis of the Y -space, without loss of generality the regression slope can be expressed as In matrix form, β$_{0}$ ( s, t ) = φ ( s ) $^{T}$Θ$_{0}$ ψ ( t ) , where φ ( s ) = ( φ$_{1}$ ( s ) , . . . , φ$_{p}$ ( s )) T and ψ ( t ) = ( ψ$_{1}$ ( t ) , . . . , ψ$_{q}$ ( t )) $^{T}$. If we also collect the component scores { U$_{k}$ } and { V$_{l}$ } into vectors U ∈ R p and V ∈ R $^{q}$, from (3), (4), (6) and (7) we obtain where W ∈ R q is the random vector with elements W$_{l}$ = 〈 Z, ψ$_{l}$ 〉 . This reduces the functional regression model (6) to a simpler multivariate regression model, and the problem now is to estimate the regression matrix Θ$_{0}$ . 5 V = Θ T $_{0}$U + W , (8) ψ ( t ) $^{T}$V = ∫ b a ψ ( t ) $^{T}$ΘT 0 φ ( s ) φ ( s ) $^{T}$U ds + Z ( t ) = ψ ( t ) $^{T}$ΘT $_{0}$U + ψ ( t ) $^{T}$W , β$_{0}$ ( s, t ) = p ∑ k =1 q ∑ l =1 θ$_{0}$$_{kl}$φ$_{k}$ ( s ) ψ$_{l}$ ( t ) . (7) Y ( t ) = µ$_{Y}$ ( t ) + ∫ b a β$_{0}$ ( s, t ) { X ( s ) − µ$_{X}$ ( s ) } ds + Z ( t ) . (6)