Estimate impulse responses via recursive (Cholesky) local projections

Estimates impulse response functions (IRFs) using recursive zero restrictions combined with local projections (Jorda, 2005). The shock of dimension e is identified by regressing the outcome at horizon h by OLS on the contemporaneous value of the e-th variable in y, controlling for the contemporaneous values of variables 1, ..., e - 1 in y, lags of the information set, and deterministic terms. This is the local-projection analogue of a Cholesky decomposition and serves as a benchmark for the heteroskedasticity-based estimator hetiv().

Usage

recols(
  y,
  O,
  X = NULL,
  Ind,
  P,
  H,
  E = 1,
  norm = 1,
  cum = FALSE,
  cov_type = "HC3",
  hpredict = 0,
  pooled = FALSE,
  details = FALSE
)

Arguments

y

Numeric matrix of stationary outcome variables (T x N). The effect on the e-th variable in each dimension e is normalized to norm at horizon 0. The first E variables are also used to impose the recursive zero restrictions.

O

Numeric matrix of information set variables (T x M). May be identical to y. Included as lags 1 through P.

X

Numeric matrix of deterministic variables (T x K). For example, time trend, seasonal dummies or other deterministic controls. Included as is (no lags). A constant is included by default.

Ind

Integer vector of length T, event indicator:

0 Control day (no event)
1 Policy day (event)
2 Contaminated control day (excluded from estimation)

At least one policy day is required.

P

Integer. Maximum lag order for the information set. Set to 0 for no lags (regression on deterministic terms only).

H

Integer. Maximum horizon (in periods) up to which IRFs are estimated.

E

Integer. Number of shock dimensions to identify via recursive ordering.

norm

Numeric scalar. Normalize the impact response of the e-th variable to shock e to a specific value. Set to 1 for standard unit-effect normalization.

cum

Logical vector of length N. For each variable in y, whether to report the cumulative impulse response instead of the level response. If only one provided, applied to all impulse responses.

cov_type

Covariance estimator for local-projection standard errors: "HC3" (default) for heteroskedasticity-robust standard errors or "NW" for Newey-West HAC standard errors. "HC3" is the default because Montiel Olea et al. (2025) show that heteroskedasticity-robust standard errors suffice for local-projection impulse responses under weak conditions, even though multi-step forecast errors are typically serially correlated. "NW" remains available as an optional HAC robustness check.

hpredict

Integer. Horizon of the responses and forecast errors used for shock prediction, on the same scale as the row labels of irf. Defaults to 0 (impact response and one-step-ahead residual). Must lie between 0 and H - 1.

pooled

Logical. If FALSE (default), the shock prediction uses the forecast errors and responses at horizon hpredict only. If TRUE, the forecast errors of all horizons 0, ..., hpredict are stacked and used jointly (pooled prediction), which weights every horizon by its signal-to-noise ratio; see kfpredict(). With hpredict = 0 both options coincide.

details

Logical. If TRUE, saves detailed OLS results and extracts shocks, which is slightly slower. If FALSE, returns only impulse responses and standard errors (e.g. for bootstrap).

Value

A named list. If details = FALSE, it contains irf, se, and Method. If details = TRUE, it contains:

irf: Array (H x N x E) of estimated impulse responses.
se: Array (H x N x E) of local-projection standard errors.
Shocks: Output of kfpredict(): predicted structural shocks.
OLSRes: List of lm model objects, one per horizon, variable, and shock dimension.
Obs: Data frame with observation counts: Tp (policy days), Tc (control days), To (contaminated days), Tt (total used).
Method: Character string "Recursive-OLS".
et: Matrix of forecast errors on event days used for shock prediction, one column per variable and horizon (y1.h0, ..., yN.h0, y1.h1, ...): horizon hpredict only if pooled = FALSE, horizons 0, ..., hpredict if pooled = TRUE.
Sig: Covariance matrix of et on event days.
SigR: Covariance matrix of the same residuals on control days, or NA if unavailable. A control day enters horizon h only if no event occurs in its window t + 1, ..., t + h.
Psi: Response matrix used for shock prediction, with rows matching the columns of et: irf[hpredict + 1, , ] (N x E) if pooled = FALSE, the rows 1, ..., hpredict + 1 of irf stacked if pooled = TRUE. With the default hpredict = 0 this is the impact response.

Details

For E > 1, identification is recursive and order-dependent: the column order of y defines both the shock ordering and the normalization variable for each shock dimension. By construction, the impact response of variable e to shock e equals norm, and the impact responses of variables 1, ..., e - 1 to shock e are zero.

Unlike hetiv(), the distinction between policy days (Ind == 1) and control days (Ind == 0) does not affect the IRF estimates: both enter the local projections, and only contaminated days (Ind == 2) are dropped. The distinction matters only for the shock extraction returned when details = TRUE.

References

Jorda, O. (2005). Estimation and inference of impulse responses by local projections. American Economic Review, 95(1), 161-182.

Montiel Olea, J. L., M. Plagborg-Moller, E. Qian, and C. K. Wolf (2025). Local projections or VARs? A primer for macroeconomists. NBER Working Paper No. 33871.

Plagborg-Moller, M. and C. K. Wolf (2021). Local projections and VARs estimate the same impulse responses. Econometrica, 89(2), 955-980.

Examples

set.seed(1)
y <- matrix(rnorm(80), ncol = 2)
Ind <- rep(0L, nrow(y))
Ind[seq(5, nrow(y), by = 5)] <- 1L
res <- recols(y = y, O = y, Ind = Ind, P = 1, H = 3, E = 2)
res$irf[1, , ] # impact responses: unit diagonal, zero above it
#>           [,1]         [,2]
#> [1,] 1.0000000 -4.26246e-17
#> [2,] 0.2783758  1.00000e+00