linear_IRF: Estimate linear impulse response function based on a single regime of a structural STVAR model.

Description

linear_IRF estimates linear impulse response function based on a single regime of a structural STVAR model.

Usage

linear_IRF(
  stvar,
  N = 30,
  regime = 1,
  which_cumulative = numeric(0),
  scale = NULL,
  ci = NULL,
  bootstrap_reps = 100,
  ncores = 2,
  robust_method = c("Nelder-Mead", "SANN", "none"),
  maxit_robust = 1000,
  seed = NULL,
  ...
)
# S3 method for irf
plot(x, shocks_to_plot, ...)
# S3 method for irf
print(x, ..., digits = 2, N_to_print, shocks_to_print)

Value

Returns a class 'irf' list with with the following elements:

$point_est:: a 3D array [variables, shock, horizon] containing the point estimates of the IRFs. Note that the first slice is for the impact responses and the slice i+1 for the period i. The response of the variable 'i1' to the shock 'i2' is subsetted as $point_est[i1, i2, ].
$conf_ints:: bootstrapped confidence intervals for the IRFs in a [variables, shock, horizon, bound] 4D array. The lower bound is obtained as $conf_ints[, , , 1], and similarly the upper bound as $conf_ints[, , , 2]. The subsetted 3D array is then the bound in a form similar to $point_est.
$all_bootstrap_reps:: IRFs from all of the bootstrap replications in a [variables, shock, horizon, rep]. 4D array. The IRF from replication i1 is obtained as $all_bootstrap_reps[, , , i1], and the subsetted 3D array is then the in a form similar to $point_est.
Other elements:: contains some of the arguments the linear_IRF was called with.

Arguments

stvar: an object of class 'stvar' defining a structural or reduced form STVAR model. For a reduced form model, the shocks are automatically identified by the lower triangular Cholesky decomposition.
N: a positive integer specifying the horizon how far ahead should the linear impulse responses be calculated.
regime: Based on which regime the linear IRF should be calculated? An integer in $1,...,M$.
which_cumulative: a numeric vector with values in $1,...,d$ (d=ncol(data)) specifying which the variables for which the linear impulse responses should be cumulative. Default is none.
scale: should the linear IRFs to some of the shocks be scaled so that they correspond to a specific instantaneous response of some specific variable? Provide a length three vector where the shock of interest is given in the first element (an integer in $1,...,d$), the variable of interest is given in the second element (an integer in $1,...,d$), and its instantaneous response in the third element (a non-zero real number). If the linear IRFs of multiple shocks should be scaled, provide a matrix which has one column for each of the shocks with the columns being the length three vectors described above.
ci: a real number in $(0, 1)$ specifying the confidence level of the confidence intervals calculated via a fixed-design wild residual bootstrap method. Available only for models that impose linear autoregressive dynamics (excluding changes in the volatility regime).
bootstrap_reps: the number of bootstrap repetitions for estimating confidence bounds.
ncores: the number of CPU cores to be used in parallel computing when bootstrapping confidence bounds.
robust_method: Should some robust estimation method be used in the estimation before switching to the gradient based variable metric algorithm? See details.
maxit_robust: the maximum number of iterations on the first phase robust estimation, if employed.
seed: a real number initializing the seed for the random generator.
...: currently not used.
x: object of class 'irf' generated by the function linear_IRF.
shocks_to_plot: IRFs of which shocks should be plotted? A numeric vector with elements in 1,...,d.
digits: the number of decimals to print
N_to_print: an integer specifying the horizon how far to print the estimates and confidence intervals. The default is that all the values are printed.
shocks_to_print: the responses to which should should be printed? A numeric vector with elements in 1,...,d. The default is that responses to all the shocks are printed.

Functions

plot(irf): plot method
print(irf): print method

Details

If the autoregressive dynamics of the model are linear (i.e., either M == 1 or mean and AR parameters are constrained identical across the regimes), confidence bounds can be calculated based on a fixed-design wild residual bootstrap method. We employ the method described in Herwartz and Lütkepohl (2014); see also the relevant chapters in Kilian and Lütkepohl (2017).

Employs the estimation function optim from the package stats that implements the optimization algorithms. The robust optimization method Nelder-Mead is much faster than SANN but can get stuck at a local solution. See ?optim and the references therein for further details.

For model identified by non-Gaussianity, the signs and ordering of the shocks are normalized by assuming that the first non-zero element of each column of the impact matrix of Regime 1 is strictly positive and they are in a decreasing order. Use the argument scale to obtain IRFs scaled for specific impact responses.

References

Herwartz H. and Lütkepohl H. 2014. Structural vector autoregressions with Markov switching: Combining conventional with statistical identification of shocks. Journal of Econometrics, 183, pp. 104-116.
Kilian L. and Lütkepohl H. 2017. Structural Vectors Autoregressive Analysis. Cambridge University Press, Cambridge.

Examples

Run this code

# \donttest{
## These are long running examples that take approximately 10 seconds to run.
## A small number of bootstrap replications is used below to shorten the
## running time (in practice, a larger number of replications should be used).

# p=1, M=1, d=2, linear VAR model with independent Student's t shocks identified
# by non-Gaussianity (arbitrary weight function applied here):
theta_112it <- c(0.644, 0.065, 0.291, 0.021, -0.124, 0.884, 0.717, 0.105, 0.322,
  -0.25, 4.413, 3.912)
mod112 <- STVAR(data=gdpdef, p=1, M=1, params=theta_112it, cond_dist="ind_Student",
 identification="non-Gaussianity", weight_function="threshold", weightfun_pars=c(1, 1))
mod112 <- swap_B_signs(mod112, which_to_swap=1:2)

# Estimate IRFs 20 periods ahead, bootstrapped 90% confidence bounds based on
# 10 bootstrap replications. Linear model so robust estimation methods are
# not required.
irf1 <- linear_IRF(stvar=mod112, N=20, regime=1, ci=0.90, bootstrap_reps=1,
 robust_method="none", seed=1, ncores=1)
plot(irf1)
print(irf1, digits=3)

# p=1, M=2, d=2, Gaussian STVAR with relative dens weight function,
# shocks identified recursively.
theta_122relg <- c(0.734054, 0.225598, 0.705744, 0.187897, 0.259626, -0.000863,
  -0.3124, 0.505251, 0.298483, 0.030096, -0.176925, 0.838898, 0.310863, 0.007512,
  0.018244, 0.949533, -0.016941, 0.121403, 0.573269)
mod122 <- STVAR(data=gdpdef, p=1, M=2, params=theta_122relg, identification="recursive")

# Estimate IRF based on the first regime 30 period ahead. Scale IRFs so that
# the instantaneous response of the first variable to the first shock is 0.3,
# and the response of the second variable to the second shock is 0.5.
# response of the Confidence bounds
# are not available since the autoregressive dynamics are nonlinear.
irf2 <- linear_IRF(stvar=mod122, N=30, regime=1, scale=cbind(c(1, 1, 0.3), c(2, 2, 0.5)))
plot(irf2)

 # Estimate IRF based on the second regime without scaling the IRFs:
irf3 <- linear_IRF(stvar=mod122, N=30, regime=2)
plot(irf3)

# p=3, M=2, d=3, Students't logistic STVAR model with the first lag of the second
# variable as the switching variable. Autoregressive dynamics restricted linear,
# but the volatility regime varies in time, allowing the shocks to be identified
# by conditional heteroskedasticity.
theta_322 <- c(0.7575, 0.6675, 0.2634, 0.031, -0.007, 0.5468, 0.2508, 0.0217, -0.0356,
 0.171, -0.083, 0.0111, -0.1089, 0.1987, 0.2181, -0.1685, 0.5486, 0.0774, 5.9398, 3.6945,
 1.2216, 8.0716, 8.9718)
mod322 <- STVAR(data=gdpdef, p=3, M=2, params=theta_322, weight_function="logistic",
  weightfun_pars=c(2, 1), cond_dist="Student", mean_constraints=list(1:2),
  AR_constraints=rbind(diag(3*2^2), diag(3*2^2)), identification="heteroskedasticity",
  parametrization="mean")

## Estimate IRFs 30 periods ahead, bootstrapped 90% confidence bounds based on
# 10 bootstrap replications. Responses of the second variable are accumulated.
irf4 <- linear_IRF(stvar=mod322, N=30, regime=1, ci=0.90, bootstrap_reps=10,
 which_cumulative=2, seed=1)
plot(irf4)
# }

Run the code above in your browser using DataLab