Local polynomials
Symmetric filters
We consider the kernel K
Asymmetric filters
Description
At the end of the period, the series is modelled as follows:
\[y = U \gamma + Z \delta + \epsilon, \: \epsilon \sim N\left(0, D\right)\]Usually, $U$ and $Z$ are parts of the polynomial matrix $X$
Constraints on the asymmetric weights are imposed by the Matrix $U=\begin{pmatrix} U_p \ U_f \end{pmatrix}$:
\[U'_p v = U' w\]The system should impose at least that the sum of the weights is equal to $1$ (which means that $U$ should contain the vector of ones).
The solution can be expressed as:
\[v=w_p + Lw_f +M w_f\] \[Lw_f =D_p^{-1} U_p\left(U_p'D_p^{-1}U_p \right)^{-1} U_f'w_f\] \[M w_f = RZ_p \delta \delta' \left( I+ Z_p' RZ_p \delta \delta'\right)^{-1} \left(Z_f'w_f-Z_p D_p^{-1}U_p\left( U_p'D_p^{-1}U_p\right)^{-1} U_f'w_f \right)\]or
\[M w_f = RZ_p \delta \delta' \left( I+ Z_p' RZ_p \delta \delta'\right)^{-1} \left(Z_f'w_f-Z_p L w_f \right)\] \[R = D_p^{-1} - D_p^{-1} U_p\left(U_p'D_p^{-1}U_p \right)^{-1} U_p'D_p^{-1}\]\(L, M \sim p \times f\) \(R \sim p \times p\)
Computation
- $t_1= D_p^{-1} U_p$ $\sim p \times u$
- $t_2=U_p’t_1 = U_p’ D_p^{-1} U_p$ $\sim u \times u$
- $t_3=U_f’w_f$ $\sim u \times 1$
- $t_4=t_2^{-1} t_3= \left(U_p’D_p^{-1}U_p \right)^{-1} U_f’ w_f$ $\sim u \times 1$
- $t_5=t_1 t_4 =D_p^{-1} U_p \left(U_p’D_p^{-1}U_p \right)^{-1} U_f’ w_f$ $\sim p \times 1$
- $t_6=t_1 t_2^{-1} t_1’=D_p^{-1} U_p\left(U_p’D_p^{-1}U_p \right)^{-1} U_p’ D_p^{-1}$ $\sim p \times p$
- $R=D_p^{-1}-t_6$ $\sim p \times p$
- $d_2=\delta \delta’$ $\sim z \times z$
- $r_1=Z_p d_2=Z_p \delta \delta’$ $\sim p \times z$
- $r_2=R r_1$ $\sim p \times z$
- $r_3=Z_p’ r_2 = Z_p’R Z_p \delta \delta’$ $\sim z \times z$