# Estimate linear impulse response function based on a single regime of a structural STVAR model. `linear_IRF` estimates linear impulse response function based on a single regime of a structural STVAR model. ```r 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 class 'irf' plot(x, shocks_to_plot, ...) ## S3 method for class 'irf' print(x, ..., digits = 2, N_to_print, shocks_to_print) ``` ## 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. ## Returns 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. ## 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. ## Functions * `plot(irf)`: plot method * `print(irf)`: print method ## Examples ```r ## 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) ``` ## 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. ## See Also `GIRF`, `GFEVD`, `fitSTVAR`, `STVAR`, `reorder_B_columns`, `swap_B_signs`

linear_IRF function