Extract monetary policy shocks via Kalman filter

Extracts structural shocks from reduced-form residuals using the Kalman filter formula. Given the response matrix Psi and the covariance matrices of residuals on event and control days, the function recovers the latent shocks for each observation. Optionally rescales shocks to unit variance.

Usage

kfpredict(Sig, SigR, Psi, et, tol = sqrt(.Machine$double.eps), scale = TRUE)

Arguments

Sig: Numeric matrix (M x M, with M = N for one horizon). Covariance matrix of the residuals in et on policy event days (e.g. Sig from hetiv()).
SigR: Numeric matrix (M x M). Covariance matrix of the residuals on control (non-event) days. Used to back out the implied shock variances when scale = TRUE.
Psi: Numeric matrix (M x E) of responses of the residuals in et to the E structural shocks (e.g. Psi from hetiv()), or an array N x E x M of responses at M horizons, which is stacked by rows.
et: Numeric matrix or data frame (T x M) of forecast errors on event days (e.g. et from hetiv()), or an array T x N x M of forecast errors at M horizons, which is stacked by columns. Rows correspond to time periods.
tol: Numeric scalar. Tolerance for the generalized inverse (MASS::ginv()). Clipped from below at sqrt(.Machine$double.eps).
scale: Logical. If TRUE (default), shocks are rescaled to unit variance using the implied shock variances recovered from Sig and SigR. This scaling assumes that structural shock variances are diagonal, so that Sig - SigR = sum_i sigma_i Psi_i Psi_i'. If FALSE, the raw Kalman filter projection is returned.

Value

A numeric matrix (T x E) of extracted structural shocks, with the same number of rows as et and one column per shock dimension. Rows with missing residuals give missing shocks.

Details

The prediction is the linear projection of the shocks on the residuals, eps_t = Sig_eps Psi' Sig^{-1} e_t, up to the scaling described under scale. It applies to any set of forecast errors that is linear in the shocks:

One horizon: Psi is the N x E response at horizon h and et holds the T x N forecast errors of y_{t+h} given information dated t - 1. With h = 0 these are the impact responses and the one-step residuals.
Pooled horizons: the forecast errors of several horizons are stacked, Psi has one block of N rows per horizon and et one block of N columns per horizon, in the same order. The pooled prediction weights every horizon by its signal-to-noise ratio and accounts for the correlation of the forecast errors across horizons. It has a weakly smaller mean squared error than the prediction from any single horizon, and coincides with the impact prediction when all responses are complete on impact.

For convenience, Psi may be supplied as an array N x E x M and et as an array T x N x M (one slice per horizon); both are then stacked internally. Sig and SigR must be of dimension N*M in that case.

The pooled prediction requires a nonsingular covariance of the stacked forecast errors. A sufficient condition is that at least as many non-policy shocks as variables drive the data (the one-day non-policy covariance has full rank). If fewer shocks are present, as in a stylised simulation with fewer regular shocks than variables, Sig is singular for stacked horizons, and the generalized inverse yields a regularized prediction that is sensitive to estimation error in Psi. A warning is issued when Sig is (nearly) singular, including when the number of complete rows of et does not exceed its number of columns.

References

Burri, M. and Kaufmann, D. (2026a). Measuring monetary policy shocks. IRENE Working Papers 24-03, IRENE Institute of Economic Research, University of Neuchatel.

Examples

Sig <- diag(2)
SigR <- diag(c(0.5, 0.5))
Psi <- matrix(c(1, 0.5), nrow = 2)
et <- matrix(rnorm(20), ncol = 2)
kfpredict(Sig = Sig, SigR = SigR, Psi = Psi, et = et)
#>              [,1]
#>  [1,] -0.53121758
#>  [2,]  0.29646687
#>  [3,]  1.11203917
#>  [4,] -0.74216125
#>  [5,]  0.87744535
#>  [6,]  0.38718522
#>  [7,]  0.26866947
#>  [8,] -0.37366525
#>  [9,] -0.15323607
#> [10,]  0.01922568

# Pooled prediction from two horizons (arrays with one slice per horizon)
Psi2 <- array(c(Psi, 0.5 * Psi), dim = c(2, 1, 2))
et2 <- array(rnorm(40), dim = c(10, 2, 2))
et2m <- matrix(et2, nrow = 10)
kfpredict(Sig = var(et2m), SigR = 0.5 * var(et2m), Psi = Psi2, et = et2)
#>              [,1]
#>  [1,] -1.36321491
#>  [2,]  0.38066380
#>  [3,]  0.52225505
#>  [4,] -1.64057001
#>  [5,] -0.54916223
#>  [6,]  0.61859522
#>  [7,]  0.06926866
#>  [8,]  0.11918553
#>  [9,] -0.37946344
#> [10,]  0.94648219