kalman_filter_skewed: Skewed Kalman Filter for Skew-Normal Structural Shocks

In dsge: Dynamic Stochastic General Equilibrium Models

Skewed Kalman Filter for Skew-Normal Structural Shocks

Description

Evaluates the log-likelihood of a linear state-space DSGE model whose structural shocks are skew-normal rather than Gaussian. The mean and covariance recursions are the usual Kalman ones (which remain the optimal linear filter under any shock distribution); in addition the filter propagates the third cumulant of the state exactly, and uses it to build a skewed one-step-ahead predictive density.

Usage

kalman_filter_skewed(y, G, H, M, D, shock_skew)

Arguments

y	Matrix of observed data (T x n_obs).
G	Policy matrix mapping states to controls.
H	State transition matrix.
M	Shock impact matrix (already scaled by the shock standard deviations, as returned by solve_dsge).
D	Observation selection matrix; the observation equation is y_t = D G x_t.
shock_skew	Numeric vector of shock skewness coefficients, one per column of M, each in (-0.99, 0.99). A value of 0 means that shock is Gaussian. A scalar is recycled across shocks.

Details

What is exact and what is approximate

The filtered and predicted means, covariances, and third cumulants are computed exactly for the linear system: the Kalman gain is the exact minimum-MSE linear filter, and the third cumulant obeys the exact linear-map rule. The only approximation is in the shape of the predictive density used for the likelihood, which is taken to be a product of univariate skew-normals in the Cholesky-decorrelated space, each matched to the exact marginal skewness. This is exact when the shocks are Gaussian and is a third-order-accurate approximation otherwise.

Skew-normal parameterisation

Each shock is standardised to zero mean and unit variance, so shock_skew is the coefficient of skewness. The largest magnitude attainable by a skew-normal is about 0.995; values are clamped to \pm 0.99.

Value

A list with the same components as the Gaussian filter – loglik, filtered_states, predicted_states, prediction_errors, filtered_P, innovation_var, predicted_obs – plus:

innovation_skew

(T x n_obs) matrix of model-implied skewness coefficients of the decorrelated one-step-ahead forecast errors.

shock_skew

The skewness vector used.

See Also

particle_filter for a fully nonlinear / non-Gaussian alternative.

Examples

m <- dsge_model(
  obs(y ~ z),
  state(z ~ rho * z),
  fixed = list(rho = 0.8))
sol <- solve_dsge(m, params = c(rho = 0.8), shock_sd = c(z = 1))
set.seed(1)
dat <- matrix(rnorm(100), 100, 1, dimnames = list(NULL, "y"))
# Gaussian shocks: identical to the standard filter
kalman_filter_skewed(dat, sol$G, sol$H, sol$M, sol$D,
                     shock_skew = 0)$loglik
# Left-skewed shocks
kalman_filter_skewed(dat, sol$G, sol$H, sol$M, sol$D,
                     shock_skew = -0.7)$loglik

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