Granger.inference.conditional: Inference on conditional Granger-causality
In grangers: Inference on Granger-Causality in the Frequency Domain

Description Usage Arguments Details Value Author(s) References See Also Examples

Inference on the conditional Granger-causality spectrum is provided generating bootstrap time series by the stationary boostrap of Politis and Romano (1994). For computational details we refer to Ding et al. (2006) and Farne' and Montanari (2018).

Granger.inference.conditional(x, y, z, ic.chosen = "SC",
  max.lag = min(4, length(x) - 1), plot = F, type.chosen = "none",
  p1 = 0, p2 = 0, nboots = 1000, conf = 0.95, bp = NULL,
  ts_boot = 1)

`x`	univariate time series.
`y`	univariate time series (of the same length of `x`).
`z`	univariate time series (of the same length of `x`).
`ic.chosen`	estimation method parameter `ic` to be passed to function VAR of package vars. Defaults to ”SC” (Schwarz criterion). Alternatives are `c(''AIC'',''HQ'',''SC'',''FPE'')`.
`max.lag`	maximum number of lags `lag.max` to be passed to function `VAR`. Defaults to `min(4, length(x) - 1)`.
`plot`	logical; if TRUE, it returns the plot of unconditional Granger-causality spectra on both directions with computed thresholds. Defaults to FALSE.
`type.chosen`	parameter `type` to be passed to function `VAR`. Defaults to `''none''`. Alternatives are `c(''none'',''const'',''trend'')`.
`p1`	parameter `p` to be passed to function VAR. It corresponds to the number of lags of the first VAR model. Defaults to 0.
`p2`	parameter `p` to be passed to function VAR. It corresponds to the number of lags of the second VAR model. Defaults to 0.
`nboots`	number of bootstrap series to be computed by function `tsbootstrap` of package tseries. It defaults to 1000.
`conf`	prescribed confidence level. It defaults to 0.95.
`bp`	matrix containing previously simulated bootstrap series, having as rows time points, as columns variables `x` and `y` (in this order). It defaults to NULL.
`ts_boot`	boolean equal to 1 if the stationary bootstrap of Politis and Romano (1994) is applied, 0 otherwise. It defaults to 1.

Granger.inference.conditional provides bootstrap inference for the Granger-causality conditional spectrum of a time series x (effect variable) on a time series z (conditioning variable) respect to a time series y (cause variable). It requires packages vars and tseries.

frequency: frequencies used by Fast Fourier Transform.

n: time series length.

nboots: number of bootstrap series used.

confidence_level: prescribed confidence level.

stat_yes: boolean equal to 0 if no stationary VAR is estimated across bootstrap samples, 1 otherwise.

non_stationarity_rate_1: percentage of non-stationary VAR models (at least one root larger than one) estimated on bootstrapped x and z.

non_stationarity_rate_2: percentage of non-stationary VAR models (at least one root larger than one) estimated on bootstrapped x and y and z.

delay1_mean: mean number of delays of stationary VAR models estimated on x and z.

delay2_mean: mean number of delays of stationary VAR models estimated on x and y and z.

quantile_conditional_causality_y.to.x.on.z: computed quantile of the Granger- causality conditional spectrum from y to x on z. Differently from function Granger.inference.unconditional, this function provides only the quantile of the unidirectional causality from y to x.

freq_y.to.x.on.z: frequencies at which the Granger-causality conditional spectrum from y to x condtional on z exceeds the computed threshold.

q_max_x.on.z: computed quantile of the Granger- causality conditional spectrum from y to x on z under Bonferroni correction. Differently from function Granger.inference.unconditional, this function provides only the quantile of the unidirectional causality from y to x.

freq_max_y.to.x.on.z: frequencies at which the Granger-causality conditional spectrum from y to x conditional on z exceeds the computed threshold under Bonferroni correction.

The result is returned invisibly if plot is TRUE.

Matteo Farne', Angela Montanari, matteo.farne2@unibo.it

Politis D. N. and Romano J. P., (1994). ”The Stationary Bootstrap”. Journal of the American Statistical Association, 89, 1303–1313.

Ding, M., Chen, Y., Bressler, S.L., 2006. Granger Causality: Basic Theory and Application to Neuroscience, Chap.17. Handbook of Time Series Analysis Recent Theoretical Developments and Applications.

Farne', M., Montanari, A., 2018. A bootstrap test to detect prominent Granger-causalities across frequencies. <arXiv:1803.00374>, Submitted.

VAR and tsbootstrap.

	RealGdp.rate.ts<-euro_area_indicators[,1]
m3.rate.ts<-euro_area_indicators[,2]
hicp.rate.ts<-euro_area_indicators[,4]
	inf_cond_m3.to.gdp.by.hicp_0.95<-
Granger.inference.conditional(RealGdp.rate.ts,m3.rate.ts,hicp.rate.ts,nboots=10)