pac_target_loading: Closed-Form Loading of a PAC Forward Sum on a Linear State...

In dsge: Dynamic Stochastic General Equilibrium Models

Closed-Form Loading of a PAC Forward Sum on a Linear State Process

Description

Collapses the infinite forward sum \sum_j w_j E_t y^*_{t+j} of a PAC equation into a single loading vector, for the common case where the target is a linear function of a state vector that evolves as s_{t+1} = \Psi s_t. Because E_t y^*_{t+j} = e' \Psi^j s_t, the sum has the exact closed form

\sum_j w_j E_t y^*_{t+j} = c\, e' \prod_k (I - \beta \phi_k \Psi)^{-1} s_t,

which avoids any truncation.

Usage

pac_target_loading(pac, Psi, e)

Arguments

pac	A "dsge_pac" object from pac_weights.
Psi	Square transition matrix \Psi of the target's state process. Must be stable (spectral radius below 1/(\beta\phi) for every root, which in practice means below one).
e	Numeric selector vector with length(e) == nrow(Psi) such that y^*_t = e' s_t.

Value

A numeric vector \ell of the same length as e, such that the PAC forward sum equals \ell' s_t.

See Also

pac_weights

Examples

pac <- pac_weights(beta = 0.99, d = 1.5)
# AR(1) target with persistence 0.8
Psi <- matrix(0.8, 1, 1)
pac_target_loading(pac, Psi, e = 1)

dsge documentation built on Sept. 25, 2026, 5:08 p.m.