Estimate impulse responses via a proxy SVAR (SVAR-IV)

Estimates a reduced-form VAR(P) by OLS, identifies the impact matrix of the first E structural shocks with external instruments (proxies) following Mertens and Ravn (2013) and Stock and Watson (2018), and computes impulse responses by recursion through the VAR. Optionally computes confidence bands with a wild bootstrap following Mertens and Ravn (2013).

Usage

svariv(
  y,
  Z,
  X = NULL,
  P,
  H,
  E = 1,
  norm = 1,
  cum = FALSE,
  nboot = 0,
  ci = c(0.9, 0.95),
  cov_type = "HC0",
  details = FALSE
)

Arguments

y: Numeric matrix of VAR variables (T x N) without missing values. The first E columns are the policy variables on which the responses are normalized.
Z: Numeric matrix of external instruments (T x M, M >= E). Missing values are allowed and drop the corresponding observation from the identification step. Set the proxy to zero (not NA) on days without an event.
X: Numeric matrix of deterministic variables (T x K), or NULL (default). Included as-is (no lags) in addition to the constant, which is always included.
P: Integer. Lag order of the VAR (P >= 1).
H: Integer. Number of horizons: IRFs are reported for horizons 0, ..., H - 1.
E: Integer. Number of shocks to identify (E < N). Default 1.
norm: Numeric scalar. Normalize the impact response of variable e to shock e to this value. Default 1.
cum: Logical scalar or vector of length N. If TRUE for variable i, the cumulative IRF is reported. A single value is recycled to all variables. Default FALSE.
nboot: Integer. Number of wild-bootstrap replications. 0 (default) skips the bootstrap, in which case se, lower, and upper are NULL.
ci: Numeric vector of confidence levels in (0, 1) for the percentile bands. Default c(0.90, 0.95).
cov_type: Covariance estimator for the robust first-stage F statistic reported with details = TRUE: "HC0" (default, Eicker-Huber-White), "HC1", "HC3", or "NW" for Newey-West HAC.
details: Logical. If TRUE, additionally returns the VAR estimates, residuals and first-stage statistics.

Value

A named list. Always contains:

irf: Array (H x N x E) of impulse responses, horizons labelled 0 to H - 1.
se: Array (H x N x E) of bootstrap standard deviations, or NULL if nboot = 0.
lower, upper: Arrays (H x N x E x length(ci)) of percentile bands, one slice per level in ci (labelled in the fourth dimension), or NULL if nboot = 0.
boot: Array (H x N x E x nboot) of bootstrap draws (NA for failed replications), or NULL if nboot = 0.
Method: Character string "SVAR-IV".

With details = TRUE, additionally contains:

Psi: Normalized impact matrix (N x E), equal to irf[1, , ] before cumulation.
Phi: Array (N x N x P) of VAR slope coefficients.
C: Vector of length N with the VAR intercepts.
D: Matrix (N x K) of coefficients on X, or NULL.
U: Matrix ((T - P) x N) of reduced-form residuals.
FStat, FStatRobust, R2: Vectors of length E with the homoskedastic first-stage F statistic, the robust effective F statistic (Montiel Olea and Pflueger, 2013) and the R-squared of the regressions of each policy residual on the proxies (see Details).
Obs: Data frame with T (sample size), Test (VAR estimation sample), and Tiv (identification sample).
nfail: Number of failed bootstrap replications.
ci: The confidence levels used for lower and upper.

Details

The first E columns of y are the policy variables: shock e is the structural shock to variable e, and the impact response of variable e to shock e is normalized to norm. For E > 1 the impact matrix of the policy block is identified up to a lower-triangular (Cholesky) rotation, so shock e has no impact effect on policy variables 1, ..., e-1; identification is therefore order-dependent, as in proxyiv() with recursive = TRUE.

All columns of Z are used jointly as instruments in the first-stage regressions of the policy residuals on the proxies. Rows of Z with missing values are excluded from the identification step only: the VAR and the reduced-form residual covariance Sigma are estimated on the full sample, as in Mertens and Ravn (2013), whereas the 2SLS step that recovers the relative impact responses uses only the observations with non-missing proxies (the identification sample).

With details = TRUE, first-stage diagnostics are reported for every policy residual: the homoskedastic F statistic of the proxies (FStat, equal to the squared t statistic when M = 1; Stock and Yogo, 2005) and the robust effective F statistic of Montiel Olea and Pflueger (2013) (FStatRobust), computed with the covariance estimator in cov_type. For E > 1 these are equation-by-equation statistics; use gweakivtest() for a joint test.

The bootstrap follows Mertens and Ravn (2013): in every replication, the VAR residuals and the proxies are multiplied by the same Rademacher draw, artificial data are generated recursively from the first P observations, and the VAR is re-estimated and re-identified. Each draw is normalized in the same way as the point estimate. Percentile bands are reported for every level in ci; se is the bootstrap standard deviation and boot holds the raw draws, so the result can be passed directly to plotirf(), which draws recentred percentile bands from the draws. Replications in which identification fails (e.g. a non-positive-definite policy block) are dropped with a warning.

References

Mertens, K. and Ravn, M. O. (2013). The dynamic effects of personal and corporate income tax changes in the United States. American Economic Review, 103(4), 1212-1247.

Montiel Olea, J. L. and Pflueger, C. (2013). A robust test for weak instruments. Journal of Business & Economic Statistics, 31(3), 358-369.

Stock, J. H. and Watson, M. W. (2018). Identification and estimation of dynamic causal effects in macroeconomics using external instruments. Economic Journal, 128(610), 917-948.

Stock, J. H. and Yogo, M. (2005). Testing for weak instruments in linear IV regression. In D. W. K. Andrews and J. H. Stock (Eds.), Identification and inference for econometric models: Essays in honor of Thomas Rothenberg (pp. 80-108). Cambridge University Press.

Examples

set.seed(1)
Phi <- array(c(0.5, 0.1, -0.1, 0.4), dim = c(2, 2, 1))
Psi <- matrix(c(1, 0.5, 0, 1), 2, 2)
Nobs <- 400
eps <- matrix(rnorm(2 * Nobs), ncol = 2)
y <- matrix(0, Nobs, 2)
for (t in 2:Nobs) y[t, ] <- Phi[, , 1] %*% y[t - 1, ] + Psi %*% eps[t, ]
Z <- matrix(eps[, 1] + rnorm(Nobs), ncol = 1)
res <- svariv(y = y, Z = Z, P = 1, H = 6, nboot = 50, ci = 0.90)
dim(res$irf)
#> [1] 6 2 1
res$irf[1, , 1] # impact responses, normalized to 1 on the first variable
#> [1] 1.0000000 0.5608632