Type:	Package
Title:	Residual-Based Fully Modified Vector Autoregression
Version:	2.0.2
Description:	Implements the Residual-Based Fully Modified Vector Autoregression (RBFM-VAR) estimator of Chang (2000) <doi:10.1017/S0266466600166071>. The RBFM-VAR procedure extends Phillips (1995) FM-VAR to handle any unknown mixture of I(0), I(1), and I(2) components without prior knowledge of the number or location of unit roots. Provides automatic lag selection via information criteria (AIC, BIC, HQ), long-run variance estimation using Bartlett, Parzen, or Quadratic Spectral kernels with Andrews (1991) <doi:10.2307/2938229> automatic bandwidth selection, Granger non-causality testing with asymptotically chi-squared Wald statistics, impulse response functions (IRF) with bootstrap confidence intervals, forecast error variance decomposition (FEVD), and out-of-sample forecasting.
License:	GPL-3
URL:	https://github.com/muhammedalkhalaf/rbfmvar
BugReports:	https://github.com/muhammedalkhalaf/rbfmvar/issues
Encoding:	UTF-8
Depends:	R (≥ 3.5.0)
Imports:	stats, MASS
Suggests:	testthat (≥ 3.0.0), knitr, rmarkdown
RoxygenNote:	7.3.3
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2026-04-02 15:26:13 UTC; SYSTEM
Author:	Muhammad Alkhalaf [aut, cre, cph], Yoosoon Chang [ctb] (Original RBFM-VAR methodology)
Maintainer:	Muhammad Alkhalaf <muhammedalkhalaf@gmail.com>
Repository:	CRAN
Date/Publication:	2026-04-09 08:20:08 UTC

rbfmvar: Residual-Based Fully Modified Vector Autoregression

Description

Implements the Residual-Based Fully Modified Vector Autoregression (RBFM-VAR) estimator following Chang (2000). The RBFM-VAR procedure extends Phillips (1995) FM-VAR to handle any unknown mixture of I(0), I(1), and I(2) components without prior knowledge of the number or location of unit roots.

Main Functions

rbfmvar

Estimate an RBFM-VAR model.

granger_test

Test for Granger non-causality.

irf

Compute impulse response functions.

fevd

Compute forecast error variance decomposition.

forecast.rbfmvar

Generate out-of-sample forecasts.

Key Features

• Handles unknown mixtures of I(0), I(1), and I(2) variables

• Automatic lag selection via AIC, BIC, or HQ

• Multiple kernels for LRV estimation (Bartlett, Parzen, QS)

• Andrews (1991) automatic bandwidth selection

• Granger non-causality testing with asymptotic chi-squared inference

• Impulse response functions with bootstrap confidence intervals

• Forecast error variance decomposition

• Out-of-sample forecasting

Methodology

The RBFM-VAR model is based on Chang (2000), which develops a fully modified VAR estimation procedure that is robust to unknown integration orders. The key innovation is using second differences to eliminate I(2) trends while applying FM corrections to handle endogeneity from I(1) regressors.

The estimator achieves:

• Zero mean mixed normal limiting distribution

• Chi-square Wald statistics for hypothesis testing

• Consistent estimation regardless of integration orders

Author(s)

Maintainer: Muhammad Alkhalaf muhammedalkhalaf@gmail.com (ORCID) [copyright holder]

Other contributors:

• Yoosoon Chang (Original RBFM-VAR methodology) [contributor]

References

Chang, Y. (2000). Vector Autoregressions with Unknown Mixtures of I(0), I(1), and I(2) Components. Econometric Theory, 16(6), 905-926. doi:10.1017/S0266466600166071

Phillips, P. C. B. (1995). Fully Modified Least Squares and Vector Autoregression. Econometrica, 63(5), 1023-1078. doi:10.2307/2171721

Andrews, D. W. K. (1991). Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation. Econometrica, 59(3), 817-858. doi:10.2307/2938229

See Also

Useful links:

• https://github.com/muhammedalkhalaf/rbfmvar

• Report bugs at https://github.com/muhammedalkhalaf/rbfmvar/issues

Extract Coefficients from rbfmvar Object

Description

Extract Coefficients from rbfmvar Object

Usage

## S3 method for class 'rbfmvar'
coef(object, type = "plus", ...)

Arguments

Value

Coefficient matrix.

Estimate Long-Run Variance Matrix

Description

Estimates the long-run variance matrix of a multivariate time series using kernel-weighted autocovariances.

Usage

estimate_lrv(v, kernel = "bartlett", bandwidth = -1, prewhiten = FALSE)

Arguments

Value

A list containing:

Omega

Estimated long-run variance matrix (n x n).

bandwidth

Bandwidth used in estimation.

kernel

Kernel used.

References

Andrews, D. W. K. (1991). Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation. Econometrica, 59(3), 817-858. doi:10.2307/2938229

Estimate One-Sided Long-Run Covariance

Description

Estimates the one-sided long-run covariance matrix.

Usage

estimate_onesided_lrv(e, v, kernel = "bartlett", bandwidth)

Arguments

Value

One-sided long-run covariance matrix (n1 x n2).

Simple VAR OLS Estimation

Description

Estimates a standard VAR model via OLS for lag selection.

Usage

estimate_var_ols(data, p)

Arguments

Value

List with residual covariance matrix.

Forecast Error Variance Decomposition

Description

Computes the forecast error variance decomposition (FEVD) from an RBFM-VAR model.

Usage

fevd(object, horizon = 20)

Arguments

Details

The FEVD shows the proportion of the forecast error variance of each variable that is attributable to shocks in each of the structural innovations. The decomposition is based on the Cholesky identification scheme, so the ordering of variables matters.

At each horizon h, the FEVD sums to 1 (100

Value

An object of class "rbfmvar_fevd" containing:

fevd

Array of FEVD values (horizon x n x n). Element [h, i, j] is the proportion of variable i's forecast error variance at horizon h explained by shocks in variable j.

horizon

FEVD horizon.

varnames

Variable names.

References

Lutkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer-Verlag. doi:10.1007/978-3-540-27752-1

Examples

# Simulate VAR data
set.seed(123)
n <- 200
e <- matrix(rnorm(n * 3), n, 3)
y <- matrix(0, n, 3)
colnames(y) <- c("y1", "y2", "y3")
for (t in 3:n) {
  y[t, ] <- 0.3 * y[t-1, ] + 0.2 * y[t-2, ] + e[t, ]
}

fit <- rbfmvar(y, lags = 2)
fv <- fevd(fit, horizon = 20)
plot(fv)

Extract Fitted Values from rbfmvar Object

Description

Extract Fitted Values from rbfmvar Object

Usage

## S3 method for class 'rbfmvar'
fitted(object, ...)

Arguments

Value

Matrix of fitted values.

Out-of-Sample Forecasting

Description

The generic function forecast computes forecasts from time series models.

Usage

forecast(object, ...)

Arguments

Value

Depends on the method dispatched. See forecast.rbfmvar for the RBFM-VAR method, which returns an object of class "rbfmvar_forecast".

Out-of-Sample Forecasting for RBFM-VAR

Description

Generates out-of-sample forecasts from an RBFM-VAR model.

Usage

## S3 method for class 'rbfmvar'
forecast(object, h = 10, level = 95, ...)

Arguments

Details

Forecasts are generated iteratively using the estimated VAR coefficients. Standard errors are computed assuming normally distributed innovations.

Note that since the RBFM-VAR is estimated on second differences, forecasts are for \Delta^2 y_{t+h}, which need to be accumulated to obtain level forecasts.

Value

An object of class "rbfmvar_forecast" containing:

mean

Matrix of point forecasts (n x h).

se

Matrix of forecast standard errors (n x h).

lower

Matrix of lower prediction bounds (n x h).

upper

Matrix of upper prediction bounds (n x h).

horizon

Forecast horizon.

level

Confidence level.

varnames

Variable names.

Examples

# Simulate VAR data
set.seed(123)
n <- 200
e <- matrix(rnorm(n * 3), n, 3)
y <- matrix(0, n, 3)
colnames(y) <- c("y1", "y2", "y3")
for (t in 3:n) {
  y[t, ] <- 0.3 * y[t-1, ] + 0.2 * y[t-2, ] + e[t, ]
}

fit <- rbfmvar(y, lags = 2)
fc <- forecast(fit, h = 10)
print(fc)
plot(fc)

Format Kernel Name for Display

Description

Format Kernel Name for Display

Usage

format_kernel(kernel)

Arguments

Value

Formatted kernel name.

Get Kernel Rate Constant

Description

Returns the rate constant c_gamma used in Andrews (1991) automatic bandwidth selection.

Usage

get_kernel_constant(kernel)

Arguments

Value

Rate constant.

Get Kernel Characteristic Exponent

Description

Returns the characteristic exponent q used in Andrews (1991) automatic bandwidth selection.

Usage

get_kernel_exponent(kernel)

Arguments

Value

Characteristic exponent q.

Get Kernel Function by Name

Description

Get Kernel Function by Name

Usage

get_kernel_function(kernel)

Arguments

Value

Kernel function.

Granger Causality Matrix

Description

Computes pairwise Granger causality tests for all variable pairs.

Usage

granger_matrix(object)

Arguments

Value

A matrix of p-values for all pairwise Granger causality tests. Row i, column j contains the p-value for "variable j causes variable i".

Examples

# Generate example data
set.seed(123)
n <- 100
mydata <- data.frame(x = cumsum(rnorm(n)), y = cumsum(rnorm(n)))
fit <- rbfmvar(mydata, lags = 2)
granger_matrix(fit)

Granger Non-Causality Test

Description

Tests for Granger non-causality in the RBFM-VAR framework using a modified Wald statistic. The test is asymptotically chi-squared under the null hypothesis, regardless of the integration order of the variables.

Usage

granger_test(object, cause, effect)

Arguments

Details

The Granger non-causality hypothesis is:

H_0: x \text{ does not Granger-cause } y

This is tested by examining whether the coefficients on lagged values of cause in the equation for effect are jointly zero.

Under the FM+ framework of Chang (2000), the Wald statistic has an asymptotic chi-squared distribution that provides a conservative (valid) p-value even when variables have unknown integration orders.

Value

A list of class "rbfmvar_granger" containing:

cause

Name of the causing variable.

effect

Name of the affected variable.

statistic

Modified Wald statistic.

df

Degrees of freedom.

p.value

P-value (conservative).

coefficients

Restricted coefficients being tested.

References

Chang, Y. (2000). Vector Autoregressions with Unknown Mixtures of I(0), I(1), and I(2) Components. Econometric Theory, 16(6), 905-926. doi:10.1017/S0266466600166071

Toda, H. Y., & Yamamoto, T. (1995). Statistical Inference in Vector Autoregressions with Possibly Integrated Processes. Journal of Econometrics, 66(1-2), 225-250. doi:10.1016/0304-4076(94)01616-8

Examples

# Simulate VAR data
set.seed(42)
n <- 200
e <- matrix(rnorm(n * 3), n, 3)
y <- matrix(0, n, 3)
colnames(y) <- c("x", "y", "z")
for (t in 3:n) {
  y[t, "x"] <- 0.5 * y[t-1, "x"] + e[t, 1]
  y[t, "y"] <- 0.3 * y[t-1, "y"] + 0.4 * y[t-1, "x"] + e[t, 2]
  y[t, "z"] <- 0.2 * y[t-1, "z"] + e[t, 3]
}

fit <- rbfmvar(y, lags = 2)

# Test if x Granger-causes y (should be significant)
test1 <- granger_test(fit, cause = "x", effect = "y")
print(test1)

# Test if z Granger-causes y (should not be significant)
test2 <- granger_test(fit, cause = "z", effect = "y")
print(test2)

Get Information Criteria Table

Description

Returns a table of information criteria values for different lag orders.

Usage

ic_table(object, max_lags = 8)

Arguments

Value

A data frame with AIC, BIC, and HQ values.

Examples

# Generate example data
set.seed(123)
n <- 100
mydata <- data.frame(x = cumsum(rnorm(n)), y = cumsum(rnorm(n)))
fit <- rbfmvar(mydata, lags = 2)
ic_table(fit, max_lags = 6)

Impulse Response Functions

Description

Computes orthogonalized impulse response functions (IRF) from an RBFM-VAR model with optional bootstrap confidence intervals.

Usage

irf(object, horizon = 20, ortho = TRUE, boot = 0, ci = 90, seed = NULL)

Arguments

Details

The IRF measures the response of each variable to a one-standard-deviation shock in each of the structural innovations. When ortho = TRUE, the structural shocks are identified using the Cholesky decomposition of the residual covariance matrix (recursive identification).

Bootstrap confidence intervals are computed using the recursive-design bootstrap following Kilian (1998).

Value

An object of class "rbfmvar_irf" containing:

irf

Array of IRF values (horizon x n x n). Element [h, i, j] is the response of variable i to a shock in variable j at horizon h.

irf_lower

Lower confidence bounds (if bootstrap was performed).

irf_upper

Upper confidence bounds (if bootstrap was performed).

horizon

IRF horizon.

varnames

Variable names.

ortho

Whether orthogonalized IRFs were computed.

boot

Number of bootstrap replications.

ci

Confidence level.

References

Kilian, L. (1998). Small-Sample Confidence Intervals for Impulse Response Functions. Review of Economics and Statistics, 80(2), 218-230. doi:10.1162/003465398557465

Lutkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer-Verlag. doi:10.1007/978-3-540-27752-1

Examples

# Simulate VAR data
set.seed(123)
n <- 200
e <- matrix(rnorm(n * 3), n, 3)
y <- matrix(0, n, 3)
colnames(y) <- c("y1", "y2", "y3")
for (t in 3:n) {
  y[t, ] <- 0.3 * y[t-1, ] + 0.2 * y[t-2, ] + e[t, ]
}

fit <- rbfmvar(y, lags = 2)
ir <- irf(fit, horizon = 20)
plot(ir)

# With bootstrap confidence intervals
ir_boot <- irf(fit, horizon = 20, boot = 500, ci = 95)
plot(ir_boot)

Bartlett (Newey-West) Kernel

Description

Bartlett (Newey-West) Kernel

Usage

kernel_bartlett(x)

Arguments

Value

Kernel weights.

Parzen Kernel

Description

Parzen Kernel

Usage

kernel_parzen(x)

Arguments

Value

Kernel weights.

Quadratic Spectral (QS) Kernel

Description

Quadratic Spectral (QS) Kernel

Usage

kernel_qs(x)

Arguments

Value

Kernel weights.

Kernel Functions for Long-Run Variance Estimation

Description

Kernel weight functions for heteroskedasticity and autocorrelation consistent (HAC) long-run variance estimation.

Details

These kernel functions are used in the estimation of long-run variance matrices following Andrews (1991) <doi:10.2307/2938229>. The kernels satisfy the conditions for consistent LRV estimation under weak dependence.

References

Andrews, D. W. K. (1991). Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation. Econometrica, 59(3), 817-858. doi:10.2307/2938229

Newey, W. K., & West, K. D. (1987). A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix. Econometrica, 55(3), 703-708. doi:10.2307/1913610

Long-Run Variance Estimation

Description

Functions for estimating long-run variance (LRV) matrices using kernel-based methods with automatic bandwidth selection.

References

Andrews, D. W. K. (1991). Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation. Econometrica, 59(3), 817-858. doi:10.2307/2938229

Newey, W. K., & West, K. D. (1994). Automatic Lag Selection in Covariance Matrix Estimation. Review of Economic Studies, 61(4), 631-653. doi:10.2307/2297912

Plot Method for rbfmvar_forecast Objects

Description

Plots forecasts from an RBFM-VAR model.

Usage

## S3 method for class 'rbfmvar_forecast'
plot(x, ...)

Arguments

Value

No return value, called for side effects (produces a plot).

Print Method for rbfmvar Objects

Description

Prints a summary of an RBFM-VAR estimation.

Usage

## S3 method for class 'rbfmvar'
print(x, ...)

Arguments

Value

Invisibly returns the input object.

Print Method for rbfmvar_forecast Objects

Description

Prints a summary of an RBFM-VAR forecast.

Usage

## S3 method for class 'rbfmvar_forecast'
print(x, ...)

Arguments

Value

Invisibly returns x.

Print Method for summary.rbfmvar Objects

Description

Prints detailed coefficient tables and diagnostics for an RBFM-VAR model summary.

Usage

## S3 method for class 'summary.rbfmvar'
print(x, digits = 4, ...)

Arguments

Value

Invisibly returns x.

Residual-Based Fully Modified VAR Estimation

Description

Estimates a Residual-Based Fully Modified Vector Autoregression (RBFM-VAR) model following Chang (2000). The RBFM-VAR procedure extends Phillips (1995) FM-VAR to handle any unknown mixture of I(0), I(1), and I(2) components without prior knowledge of the number or location of unit roots.

Usage

rbfmvar(
  data,
  lags = 2,
  max_lags = 8,
  ic = "none",
  kernel = "bartlett",
  bandwidth = -1,
  level = 95
)

Arguments

Details

The RBFM-VAR model is specified as:

\Delta^2 y_t = \sum_{j=1}^{p-2} \Gamma_j \Delta^2 y_{t-j} + \Pi_1 \Delta y_{t-1} + \Pi_2 y_{t-1} + e_t

where \Delta is the difference operator and \Delta^2 = \Delta \circ \Delta.

The FM+ correction eliminates the second-order asymptotic bias that arises from the correlation between the regression errors and the innovations in integrated regressors. The estimator achieves:

• Zero mean mixed normal limiting distribution

• Chi-square Wald statistics for hypothesis testing

• Robustness to unknown integration orders

Value

An object of class "rbfmvar" containing:

F_ols

OLS coefficient matrix.

F_plus

FM+ corrected coefficient matrix.

SE_mat

Standard errors for FM+ coefficients.

Pi1_ols, Pi1_plus

Coefficient matrices for \Delta y_{t-1}.

Pi2_ols, Pi2_plus

Coefficient matrices for y_{t-1}.

Gamma_ols, Gamma_plus

Coefficient matrices for \Delta^2 y_{t-j} (if p >= 3).

Sigma_e

Residual covariance matrix.

Omega_ev, Omega_vv

Long-run variance components.

Delta_vdw

One-sided long-run covariance for FM correction.

residuals

Matrix of residuals from FM+ estimation.

fitted

Matrix of fitted values.

nobs

Number of observations in original data.

T_eff

Effective sample size after differencing.

n_vars

Number of variables.

p_lags

VAR lag order used.

bandwidth

Bandwidth used for LRV estimation.

kernel

Kernel used for LRV estimation.

ic

Information criterion used (if any).

varnames

Variable names.

call

The matched call.

References

Chang, Y. (2000). Vector Autoregressions with Unknown Mixtures of I(0), I(1), and I(2) Components. Econometric Theory, 16(6), 905-926. doi:10.1017/S0266466600166071

Phillips, P. C. B. (1995). Fully Modified Least Squares and Vector Autoregression. Econometrica, 63(5), 1023-1078. doi:10.2307/2171721

Andrews, D. W. K. (1991). Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation. Econometrica, 59(3), 817-858. doi:10.2307/2938229

Examples

# Simulate a simple VAR(2) process
set.seed(123)
n <- 200
e <- matrix(rnorm(n * 3), n, 3)
y <- matrix(0, n, 3)
for (t in 3:n) {
  y[t, ] <- 0.3 * y[t-1, ] + 0.2 * y[t-2, ] + e[t, ]
}
colnames(y) <- c("y1", "y2", "y3")

# Estimate RBFM-VAR
fit <- rbfmvar(y, lags = 2)
summary(fit)

# With automatic lag selection
fit_aic <- rbfmvar(y, max_lags = 6, ic = "aic")
summary(fit_aic)

Core RBFM-VAR Estimation

Description

Core RBFM-VAR Estimation

Usage

rbfmvar_estimate(Y, p, kernel, bandwidth)

Arguments

Value

List of estimation results.

Extract Residuals from rbfmvar Object

Description

Extract Residuals from rbfmvar Object

Usage

## S3 method for class 'rbfmvar'
residuals(object, ...)

Arguments

Value

Matrix of residuals.

Andrews (1991) Automatic Bandwidth Selection

Description

Implements the data-dependent automatic bandwidth selection procedure of Andrews (1991).

Usage

select_bandwidth_andrews(v, kernel = "bartlett")

Arguments

Value

Optimal bandwidth.

References

Andrews, D. W. K. (1991). Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation. Econometrica, 59(3), 817-858. doi:10.2307/2938229

Lag Selection via Information Criteria

Description

Selects the optimal VAR lag order using information criteria (AIC, BIC, or HQ).

Usage

select_lags_ic(data, max_lags, ic = "aic")

Arguments

Value

A list containing:

best_p

Optimal lag order.

ic_table

Data frame of IC values for each lag.

References

Akaike, H. (1974). A New Look at the Statistical Model Identification. IEEE Transactions on Automatic Control, 19(6), 716-723. doi:10.1109/TAC.1974.1100705

Schwarz, G. (1978). Estimating the Dimension of a Model. The Annals of Statistics, 6(2), 461-464. doi:10.1214/aos/1176344136

Hannan, E. J., & Quinn, B. G. (1979). The Determination of the Order of an Autoregression. Journal of the Royal Statistical Society: Series B, 41(2), 190-195. doi:10.1111/j.2517-6161.1979.tb01072.x

Summary Method for rbfmvar Objects

Description

Provides detailed summary of RBFM-VAR estimation results.

Usage

## S3 method for class 'rbfmvar'
summary(object, ...)

Arguments

Value

A list of class "summary.rbfmvar" containing summary information.

Extract Variance-Covariance Matrix from rbfmvar Object

Description

Extract Variance-Covariance Matrix from rbfmvar Object

Usage

## S3 method for class 'rbfmvar'
vcov(object, ...)

Arguments

Value

Error covariance matrix.