# Compute nonlinear impulse responses Compute nonlinear impulse responses with local projections by Jordà (2005). The data can be separated into two states by a smooth transition function as applied in Auerbach and Gorodnichenko (2012), or by a simple dummy approach. ```r lp_nl( endog_data, lags_endog_lin = NULL, lags_endog_nl = NULL, lags_criterion = NaN, max_lags = NaN, trend = NULL, shock_type = NULL, confint = NULL, use_nw = TRUE, nw_lag = NULL, nw_prewhite = FALSE, adjust_se = FALSE, hor = NULL, switching = NULL, lag_switching = TRUE, use_logistic = TRUE, use_hp = NULL, lambda = NULL, gamma = NULL, exog_data = NULL, lags_exog = NULL, contemp_data = NULL, num_cores = 1 ) ``` ## Arguments - `endog_data`: A data.frame , containing all endogenous variables for the VAR. The Cholesky decomposition is based on the column order. - `lags_endog_lin`: NaN or integer. NaN if lag length criterion is used. Integer for number of lags for linear VAR to identify shock. - `lags_endog_nl`: NaN or integer. Number of lags for nonlinear VAR. NaN if lag length criterion is given. - `lags_criterion`: NaN or character. NaN (default) means that the number of lags will be given at **lags_endog_nl** and **lags_endog_lin**. The lag length criteria are 'AICc', 'AIC' and 'BIC'. - `max_lags`: NaN or integer. Maximum number of lags (if **lags_criterion** = 'AICc', 'AIC', 'BIC'). NaN (default) otherwise. - `trend`: Integer. Include no trend = 0 , include trend = 1, include trend and quadratic trend = 2. - `shock_type`: Integer. Standard deviation shock = 0, unit shock = 1. - `confint`: Double. Width of confidence bands. 68% = 1; 90% = 1.65; 95% = 1.96. - `use_nw`: Boolean. Use Newey-West (1987) standard errors for impulse responses? TRUE (default) or FALSE. - `nw_lag`: Integer. Specifies the maximum lag with positive weight for the Newey-West estimator. If set to NULL (default), the lag increases with with the number of horizon. - `nw_prewhite`: Boolean. Should the estimators be pre-whitened? TRUE of FALSE (default). - `adjust_se`: Boolen. Should a finite sample adjsutment be made to the covariance matrix estimators? TRUE or FALSE (default). - `hor`: Integer. Number of horizons for impulse responses. - `switching`: Numeric vector. A column vector with the same length as **endog_data**. If 'use_logistic = TRUE', this series can either be decomposed via the Hodrick-Prescott filter (see Auerbach and Gorodnichenko, 2013) or directly plugged into the following logistic function: $$ F_{z_t} = \frac{exp(-\gamma z_t)}{1 + exp(-\gamma z_t)}. $$ Important: $F_{z_t}$ will be lagged by one and then multiplied with the data. If the variable shall not be lagged, use 'lag_switching = FALSE': Regime 1 = (1-$F(z_{t-1})$)*y_(t-p), Regime 2 = $F(z_{t-1})$*y_(t-p). - `lag_switching`: Boolean. Use the first lag of the values of the transition function? TRUE (default) or FALSE. - `use_logistic`: Boolean. Use logistic function to separate states? TRUE (default) or FALSE. If FALSE, the values of the switching variable have to be binary (0/1). - `use_hp`: Boolean. Use HP-filter? TRUE or FALSE. - `lambda`: Double. Value of $\lambda$ for the Hodrick-Prescott filter (if use_hp = TRUE). - `gamma`: Double. Positive number which is used in the transition function. - `exog_data`: A data.frame , containing exogenous variables for the VAR. The row length has to be the same as **endog_data**. Lag lengths for exogenous variables have to be given and will not be determined via a lag length criterion. - `lags_exog`: NULL or Integer. Integer for the number of lags for the exogenous data. The value cannot be 0. If you want to to include exogenous data with contemporaneous impact use **contemp_data**. - `contemp_data`: A data.frame , containing exogenous data with contemporaneous impact. This data will not be lagged. The row length has to be the same as **endog_data**. - `num_cores`: Integer. The number of cores to use for the estimation. If NULL, the function will use the maximum number of cores minus one. ## Returns A list containing: - **irf_s1_mean**: A three 3D array , containing all impulse responses for all endogenous variables of the first state. The last dimension denotes the shock variable. The row in each matrix denotes the responses of the **ith** variable, ordered as in **endog_data**. The columns are the horizons. For example, if the results are saved in **results_nl**, results_nl$irf_s1_mean[, , 1] returns a KXH matrix, where K is the number of variables and H the number of horizons. '1' is the shock variable, corresponding to the variable in the first column of **endog_data**. - **irf_s1_low**: A three 3D array , containing all lower confidence bands of the impulse responses, based on robust standard errors by Newey and West (1987). Properties are equal to **irf_s1_mean**. - **irf_s1_up**: A three 3D array , containing all upper confidence bands of the impulse responses, based on robust standard errors by Newey and West (1987). Properties are equal to **irf_s1_mean**. - **irf_s2_mean**: A three 3D array , containing all impulse responses for all endogenous variables of the second state. The last dimension denotes the shock variable. The row in each matrix denotes the responses of the **ith** variable, ordered as in endog_data. The columns denote the horizon. For example, if the results are saved in **results_nl**, results_nl$irf_s2_mean[, , 1] returns a KXH matrix, where K is the number of variables and H the number of horizons. '1' is the first shock variable corresponding to the variable in the first column of **endog_data**. - **irf_s2_low**: A three 3D array , containing all lower confidence bands of the responses, based on robust standard errors by Newey and West (1987). Properties are equal to **irf_s2_mean**. - **irf_s2_up**: A three 3D array , containing all upper confidence bands of the responses, based on robust standard errors by Newey and West (1987). Properties are equal to **irf_s2_mean**. - **specs**: A list with properties of **endog_data** for the plot function. It also contains lagged data (y_nl and x_nl) used for the irf estimations, and the selected lag lengths when an information criterion has been used. - **fz**: A vector containing the values of the transition function F(z_t-1). ## Examples ```r ## Example without exogenous variables ## # Load package library(lpirfs) library(gridExtra) library(ggpubr) # Load (endogenous) data endog_data <- interest_rules_var_data # Choose data for switching variable (here Federal Funds Rate) # Important: The switching variable does not have to be used within the VAR! switching_data <- endog_data$Infl # Estimate model and save results results_nl <- lp_nl(endog_data, lags_endog_lin = 4, lags_endog_nl = 3, trend = 0, shock_type = 1, confint = 1.96, hor = 24, switching = switching_data, use_hp = TRUE, lambda = 1600, gamma = 3) # Show all plots plot(results_nl) # Make and save all plots nl_plots <- plot_nl(results_nl) # Save plots based on states s1_plots <- sapply(nl_plots$gg_s1, ggplotGrob) s2_plots <- sapply(nl_plots$gg_s2, ggplotGrob) # Show first irf of each state plot(s1_plots[[1]]) plot(s2_plots[[1]]) # Show diagnostics. The first element correponds to the first shock variable. summary(results_nl) ## Example with exogenous variables ## # Load (endogenous) data endog_data <- interest_rules_var_data # Choose data for switching variable (here Federal Funds Rate) # Important: The switching variable does not have to be used within the VAR! switching_data <- endog_data$FF # Create exogenous data and data with contemporaneous impact (for illustration purposes only) exog_data <- endog_data$GDP_gap*endog_data$Infl*endog_data$FF + rnorm(dim(endog_data)[1]) contemp_data <- endog_data$GDP_gap*endog_data$Infl*endog_data$FF + rnorm(dim(endog_data)[1]) # Exogenous data has to be a data.frame exog_data <- data.frame(xx = exog_data) contemp_data <- data.frame(cc = contemp_data) # Estimate model and save results results_nl <- lp_nl(endog_data, lags_endog_lin = 4, lags_endog_nl = 3, trend = 0, shock_type = 1, confint = 1.96, hor = 24, switching = switching_data, use_hp = TRUE, lambda = 1600, # Ravn and Uhlig (2002): # Anuual data = 6.25 # Quarterly data = 1600 # Monthly data = 129 600 gamma = 3, exog_data = exog_data, lags_exog = 3) # Show all plots plot(results_nl) # Show diagnostics. The first element correponds to the first shock variable. summary(results_nl) ``` ## References Akaike, H. (1974). "A new look at the statistical model identification", **IEEE Transactions on Automatic Control**, 19 (6): 716–723. Auerbach, A. J., and Gorodnichenko Y. (2012). "Measuring the Output Responses to Fiscal Policy." **American Economic Journal: Economic Policy**, 4 (2): 1-27. Auerbach, A. J., and Gorodnichenko Y. (2013). "Fiscal Multipliers in Recession and Expansion." **NBER Working Paper Series**. Nr. 17447. Hurvich, C. M., and Tsai, C.-L. (1989), "Regression and time series model selection in small samples", **Biometrika**, 76(2): 297–307 Jordà, Ò. (2005) "Estimation and Inference of Impulse Responses by Local Projections." **American Economic Review**, 95 (1): 161-182. Newey, W.K., and West, K.D. (1987). “A Simple, Positive-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix.” **Econometrica**, 55, 703–708. Schwarz, Gideon E. (1978). "Estimating the dimension of a model", **Annals of Statistics**, 6 (2): 461–464. Ravn, M.O., Uhlig, H. (2002). "On Adjusting the Hodrick-Prescott Filter for the Frequency of Observations." **Review of Economics and Statistics**, 84(2), 371-376. ## See Also [https://adaemmerp.github.io/lpirfs/README_docs.html](https://adaemmerp.github.io/lpirfs/README_docs.html) ## Author(s) Philipp Adämmer

lp_nl function