# lp_nl_iv: Compute nonlinear impulse responses with identified shock In lpirfs: Local Projections Impulse Response Functions

## Description

Compute nonlinear impulse responses with local projections and identified shock. 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.

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 lp_nl_iv( endog_data, lags_endog_nl = NULL, shock = NULL, instr = NULL, exog_data = NULL, lags_exog = NULL, contemp_data = NULL, lags_criterion = NaN, max_lags = NaN, trend = 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, num_cores = 1 ) 

## Arguments

 endog_data A data.frame, containing all endogenous variables for the VAR. lags_endog_nl NaN or integer. NaN if lags are chosen by a lag length criterion. Integer for number of lags for endog_data. shock One column data.frame, including the instrument to shock with. The row length has to be the same as endog_data. instr Deprecated input name. Use shock instead. See shock for details. exog_data A data.frame, containing exogenous variables. The row length has to be the same as endog_data. Lag lengths for exogenous variables have to be given and will no be determined via a lag length criterion. lags_exog NULL or Integer. Integer for the number of lags for the exogenous 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. lags_criterion NaN or character. NaN means that the number of lags will be given at lags_endog_nl. Possible lag length criteria are 'AICc', 'AIC' or 'BIC'. max_lags NaN or integer. Maximum number of lags (if lags_criterion = 'AICc', 'AIC', 'BIC'). NaN otherwise. trend Integer. Include no trend = 0 , include trend = 1, include trend and quadratic trend = 2. 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. This series can either be decomposed via the Hodrick-Prescott filter (see Auerbach and Gorodnichenko, 2013) or directly plugged into the following smooth transition function: F_{z_t} = \frac{exp(-γ z_t)}{1 + exp(-γ z_t)}. Warning: F_{z_t} will be lagged by one and then multiplied with the data. If the variable shall not be lagged, the vector has to be given with a lead of one. The data for the two regimes are: 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 λ for the Hodrick-Prescott filter (if use_hp = TRUE). gamma Double. Positive number which is used in the transition function. 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.

## Value

A list containing:

 irf_s1_mean A matrix, containing the impulse responses of the first regime. The row in each matrix denotes the responses of the ith variable to the shock. The columns are the horizons. irf_s1_low A matrix, 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 matrix, 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 matrix, containing all impulse responses for the second regime. The row in each matrix denotes the responses of the ith variable to the shock. The columns denote the horizon. irf_s2_low A matrix, 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 matrix, 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 estimations of the impulse responses, 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).

## 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.

Blanchard, O., and Perotti, R. (2002). “An Empirical Characterization of the Dynamic Effects of Changes in Government Spending and Taxes on Output.” Quarterly Journal of Economics, 117(4): 1329–1368.

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.

Jordà, Ò, Schularick, M., Taylor, A.M. (2015), "Betting the house", Journal of International Economics, 96, S2-S18.

Newey, W.K., and West, K.D. (1987). “A Simple, Positive-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix.” Econometrica, 55, 703–708.

Ramey, V.A., and Zubairy, S. (2018). "Government Spending Multipliers in Good Times and in Bad: Evidence from US Historical Data." Journal of Political Economy, 126(2): 850 - 901.

Schwarz, Gideon E. (1978). "Estimating the dimension of a model", Annals of Statistics, 6 (2): 461–464.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 # This example replicates results from the Supplementary Appendix # by Ramey and Zubairy (2018) (RZ-18). # Load and prepare data ag_data <- ag_data sample_start <- 7 sample_end <- dim(ag_data)[1] endog_data <- ag_data[sample_start:sample_end, 3:5] # The shock is estimated by RZ-18 shock <- ag_data[sample_start:sample_end, 7] # Include four lags of the 7-quarter moving average growth rate of GDP # as exogenous variables (see RZ-18) exog_data <- ag_data[sample_start:sample_end, 6] # Use the 7-quarter moving average growth rate of GDP as switching variable # and adjust it to have suffiently long recession periods. switching_variable <- ag_data$GDP_MA[sample_start:sample_end] - 0.8 # Estimate local projections results_nl_iv <- lp_nl_iv(endog_data, lags_endog_nl = 3, shock = shock, exog_data = exog_data, lags_exog = 4, trend = 0, confint = 1.96, hor = 20, switching = switching_variable, use_hp = FALSE, gamma = 3) # Show all impulse responses plot(results_nl_iv) # Make and save individual plots plots_nl_iv <- plot_nl(results_nl_iv) # Show single impulse responses # Compare with red line of left plot (lower panel) in Figure 12 in Supplementary Appendix of RZ-18. plot(plots_nl_iv$gg_s1[[1]]) # Compare with blue line of left plot (lower panel) in Figure 12 in Supplementary Appendix of RZ-18. plot(plots_nl_iv\$gg_s2[[1]]) # Show diagnostics. The first element shows the reaction of the first endogenous variable. summary(results_nl_iv)