Package {sysbreak}

Structural Breaks in a System of Equations

Description

Estimation and testing for multivariate structural break models with coefficient breaks, covariance breaks, or both.

Main Functions

• sysbreak: Breaks in a linear system of equations, including common-regressor and SUR-style structural break models

• varbreak: VAR structural break models

• netbreak: Network VAR structural break models

• ordbreak: Ordered two-break models

• ordcheck: Coefficient layout of an ordered two-break model

• brtype: Tests of what changes at each break

• dyspill: Generalized Diebold-Yilmaz spillover measures

• dynobreak: The same spillover measures without breaks

Break Types

The package supports the following structural break specifications:

• Coefficient breaks

• Covariance breaks

• Joint coefficient and covariance breaks

Author(s)

Maintainer: Zhongjun Qu qu@bu.edu

Authors:

• Cheolju Kim cheolju@bu.edu

• Pierre Perron

References

Qu, Z., and Perron, P. (2007). Estimating and testing structural changes in multivariate regressions. Econometrica, 75(2), 459-502.

Bai, J., and Perron, P. (2003). Critical values for multiple structural change tests. The Econometrics Journal, 6(1), 72-78.

Methods to Examine What Changes at Each Break Date

Description

For each estimated break date, tests whether the volatilities, the correlations and the regression coefficients differ between the two regimes on either side of it.

Usage

brtype(fit)

Arguments

Details

The volatility statistic is a two-sample Hotelling (1931) statistic on the squared residuals. The correlation statistic is the Jennrich (1970) test for equal correlation matrices, which assumes normal errors. The coefficient statistic is a Wald test for equal coefficients; it covers only the coefficients allowed to change, and is NA when no coefficient is allowed to change. Each test compares the two regimes adjacent to a break and treats the break dates as known.

The coefficient test holds the error covariance constant across regimes, even when fit allows it to break (brv = TRUE). Estimating the covariance separately in each regime would make the test oversized in samples of typical length; the constant covariance tends to make it conservative when the volatility does change.

Value

A data frame with one row per break: breaks, the break date as a row of the data; the statistics volatility, correlation and coefficients; and their p-values volatility_p, correlation_p and coefficients_p. The degrees of freedom are n, n(n-1)/2 and the number of coefficients allowed to change.

References

Hotelling, H. (1931). The generalization of Student's ratio. The Annals of Mathematical Statistics, 2(3), 360-378.

Jennrich, R. I. (1970). An asymptotic chi-square test for the equality of two correlation matrices. Journal of the American Statistical Association, 65, 904-912.

Examples

set.seed(11)
bigt <- 120
x <- rnorm(bigt)
y1 <- 1 + 0.4 * x + rnorm(bigt, sd = 0.3)
y2 <- -0.5 + 0.2 * x + rnorm(bigt, sd = 0.3)
y1[61:120] <- y1[61:120] + 1.2 - 0.6 * x[61:120]
dat <- data.frame(y1 = y1, y2 = y2, x = x)

fit <- sysbreak(cbind(y1, y2) ~ x, data = dat, m = 2, trim = 0.2)
brtype(fit)

GAUSS/MATLAB Example Data 1

Description

Example data used by the original GAUSS/MATLAB mbreak2 script. In this package the corresponding procedure is sysbreak.

Usage

data1

Format

A numeric matrix with 150 rows and 5 columns. Columns 1 and 2 are dependent variables, and columns 3 through 5 are regressors.

Source

Qu, Z., and Perron, P. (2007). Estimating and testing structural changes in multivariate regressions. Econometrica, 75(2), 459-502.

GAUSS/MATLAB Example Data 2

Description

Example data used by the original GAUSS/MATLAB obreak2 script for the ordered-break procedure.

Usage

data2

Format

A numeric matrix with 100 rows and 5 columns. Columns 1 and 2 are dependent variables, and columns 3 through 5 are regressors.

Source

Qu, Z., and Perron, P. (2007). Estimating and testing structural changes in multivariate regressions. Econometrica, 75(2), 459-502.

Diebold-Yilmaz Spillover Measures Without Breaks

Description

Fits a VAR by OLS on the whole sample and computes the generalized spillover measures of dyspill(), for comparison with the measures of a model with breaks.

Usage

dynobreak(y, lags, h, intercept = TRUE)

Arguments

Value

A list with the row-normalized generalized fevd, the total spillover index total, and the directional vectors from, to and net. The covariance divides by the number of observations.

Examples

set.seed(1)
bigt <- 120
a <- matrix(c(0.2, 0.4, 0.4, 0.2), 2)
y <- matrix(0, bigt, 2)
colnames(y) <- c("y1", "y2")
for (tt in 2:bigt) {
  y[tt, ] <- a %*% y[tt - 1, ] + rnorm(2, sd = 0.3)
}

sp <- dynobreak(y, lags = 1, h = 10)
sp$total
sp$net

Diebold-Yilmaz Generalized Spillover Measures

Description

Computes horizon-specific spillover measures from a fitted structural-break VAR model. The forecast-error variance decomposition uses generalized shocks and is therefore invariant to variable ordering.

Usage

dyspill(fit, h = 10, boot = 0, seed = NULL)

Arguments

Details

For a regime-specific VAR,

y_t = c_r + \sum_{\ell=1}^p A_{\ell,r}y_{t-\ell}+u_t,\qquad E(u_tu_t')=\Sigma_r,

let \Psi_{0,r}=I and

\Psi_{s,r}=\sum_{\ell=1}^{\min(p,s)} A_{\ell,r}\Psi_{s-\ell,r}.

The generalized variance share from shock j to variable i is

\theta_{ij}^{g}(H)= \frac{\sigma_{jj}^{-1}\sum_{s=0}^{H-1} (e_i'\Psi_s\Sigma e_j)^2} {\sum_{s=0}^{H-1}e_i'\Psi_s\Sigma\Psi_s'e_i}.

Rows are normalized to sum to one before spillover statistics are formed. For variable i, from is the contribution received from all other shocks, to is the contribution transmitted to all other variables, and net = to - from; a positive net value identifies a net transmitter. The total spillover index is the average off-diagonal contribution, expressed in percent.

The benchmark puts every lag matrix to zero and keeps \Sigma_r. The measures then depend on the error correlations only; comparing them with the fitted ones shows how much of the spillover comes from the dynamics. With two variables the benchmark total is 100\rho^2/(1+\rho^2).

The bootstrap is a recursive residual bootstrap within each regime: the residuals of the regime are resampled, the series is rebuilt from the fitted VAR of the regime, the model is refitted at the same break dates in the same way as fit, and the measures are recomputed. The break dates are treated as known. The intervals are percentile intervals, since the draws are skewed in small samples. Draws in which a refitted VAR is not stable are dropped.

Value

An object of class "dyspill" containing:

horizon

The forecast horizon h.

model

The source model type.

variables

Names of the endogenous variables.

break_grid

A data frame with one row per regime: the regime number; its first and last observation, start and end, counted in the estimation sample (which omits the first lags observations); and the coefficient and covariance regimes it belongs to, coef_regime and cov_regime.

regimes

A list with one entry per regime. Each entry contains:

• input_rows: the first and last row of the data in the regime.

• spectral_radius: the largest modulus of the eigenvalues of the companion matrix of the VAR; the VAR is stable when it is below one.

• fevd: the generalized forecast-error variance decomposition, with rows normalized to sum to one.

• total: the total spillover index.

• from, to and net: the directional spillovers of each variable.

• net_pairwise: the matrix of net pairwise spillovers.

• benchmark_total and benchmark_net: total and net recomputed with all lag coefficients set to zero, so that they reflect the error correlations only (see Details).

• With boot > 0: the bootstrap standard errors total_se and net_se, and the percentile intervals total_ci and net_ci.

boot

Returned only when boot > 0. A list with:

• replications: the number of bootstrap replications.

• level: the coverage probability of the intervals, taken from ci_level of the fit.

• total: the bootstrap draws of the total index, a matrix with one row per replication and one column per regime.

• net: the bootstrap draws of the net spillovers, an array of replications by variables by regimes.

• differences: a data frame with one row per pair of regimes, giving the difference in the total index (the later regime minus the earlier one), its bootstrap standard error se, and the percentile interval lower and upper.

• dropped: the number of draws dropped in each regime because the refitted VAR was not stable.

References

Diebold, F. X., and Yilmaz, K. (2012). Better to give than to receive: Predictive directional measurement of volatility spillovers. International Journal of Forecasting, 28(1), 57–66.

Examples

set.seed(1)
bigt <- 120
y <- matrix(0, bigt, 2)
colnames(y) <- c("y1", "y2")
for (tt in 2:bigt) {
  a <- if (tt > 60) matrix(c(0.2, 0.6, 0.6, 0.2), 2) else diag(0.2, 2)
  y[tt, ] <- a %*% y[tt - 1, ] + rnorm(2, sd = 0.3)
}

fit <- varbreak(y, lags = 1, m = 1, trim = 0.2)
dyspill(fit, h = 10, boot = 99, seed = 1)

U.S. House Prices and the Macroeconomy

Description

Quarterly data for the house price application of Ahamada and Diaz Sanchez (2013), 1959:Q4 to 2009:Q3.

Usage

housing

Format

A data frame with 200 rows and 6 columns:

date

Quarter, such as "1960:Q1".

dGDP

Log difference of real GDP.

dC

Log difference of real personal consumption expenditures.

dHP

Log difference of the real home price index.

FFR

Federal funds rate, in percent, first month of the quarter.

dI

Change in FFR.

Source

FRED series GDPC1, PCEC, CPIAUCSL and FEDFUNDS, and the Shiller nominal home price index, seasonally adjusted by log-STL. Consumption and home prices are deflated by CPIAUCSL.

References

Ahamada, I., and Diaz Sanchez, J. L. (2013). A retrospective analysis of the house price macro-relationship in the United States. International Journal of Central Banking, 9(4), 153-174.

International Inflation

Description

Monthly consumer price inflation for the G7 countries and the euro area, 1973:04 to 2007:12, from Bataa, Osborn, Sensier and van Dijk (2013).

Usage

inflation

Format

A data frame with 417 rows and 9 columns: date, such as "1973:04", followed by inflation for Canada, EuroArea, France, Germany, Italy, Japan, UK and US. Inflation is 100 \times \Delta \log of the consumer price index, adjusted for outliers and seasonality by the authors.

Source

The replication files of Bataa et al. (2013).

References

Bataa, E., Osborn, D. R., Sensier, M., and van Dijk, D. (2013). Structural breaks in the international dynamics of inflation. Review of Economics and Statistics, 95(2), 646-659.

Structural Breaks in a network VAR

Description

Estimates structural breaks in a VAR whose coefficients are restricted by a fixed network matrix. With one lag, the model is

y_t = \mu_j + (\alpha_{1,j} W + \beta_{1,j} I)y_{t-1} + u_t,

for observations in regime j. With multiple lags, the model is

y_t = \mu_j + \sum_{\ell=1}^p (\alpha_{\ell,j} W + \beta_{\ell,j} I)y_{t-\ell} + u_t.

Usage

netbreak(y, W, lags, m, intercept = TRUE,
  coef_breaks = "all", row_normalize = FALSE,
  test_level = 0.05, ci_level = 0.95,
  trim = 0.15, verbose = FALSE, brv = FALSE,
  robust = c("none", "hc", "hac"), hetq = FALSE, prewhit = FALSE,
  restart = c("none", "brbp", "brbp_residual"),
  restart_B = 10, restart_verbose = FALSE, h = NULL)

Arguments

Details

The interpretation of testing$statistic depends on robust.

• robust = "none" gives the SupLR statistic. The code can test the parameters specified by coef_breaks, the disturbance covariance when brv = TRUE, or both.

• robust = "hc" or "hac" gives the robust SupF statistic, for the parameters only, because the covariance-break critical values are case dependent under heteroskedasticity or serial correlation; it assumes a constant error covariance. With brv = TRUE, the estimates, standard errors and confidence intervals account for the covariance breaks; under "hc" the intervals further require errors without ARCH-type dependence.

Value

An object of class c("netbreak", "sysbreak") containing:

spec

A summary of the setup, including the sample size bigt, the number of equations n, the total number of NVAR parameters per regime q, the arguments m, lags, intercept, row_normalize, trim, brv, robust, hetq, prewhit, test_level and ci_level, the network matrix W, the equation and parameter names, the retained input rows y_rows after discarding the lags, and a record of the coef_breaks selector.

testing

The tests, reported at test_level: statistic, testing no break against k breaks for every k = 1,\ldots,m, with its critical_values; statistic_type, "SupLR" or "SupF", and NA when a requested robust test is unavailable; wdmax and wdmax_critical_values; the sequential tests of l against l+1 breaks, each containing l, a statistic and its critical_values; and the number of changing_parameters, which includes the changing covariance parameters when robust = "none".

br_selected

The outcome of the break-number determination at test_level: the selected number of breaks, the WDmax statistic with its critical value and significance flag, and the sequential steps taken. breaks is 0 when WDmax is not significant, and NA when the tests are unavailable for the requested specification.

estimation

The estimates related to the selected number of breaks: estimated breaks; their confidence intervals ci_breaks; coefficients by regimes and their confidence intervals; disturbance covariance matrices. This component is NULL when no break is detected, in which case only testing and br_selected are informative.

spillover

When h is supplied and at least one break is selected, regime-specific Diebold–Yilmaz generalized spillover measures returned by dyspill.

y

The responses used in estimation, rows spec$y_rows of y.

Note that m is the maximum number of breaks allowed. Testing output is reported for every value from 1 through m, while estimation corresponds to the number of breaks selected by the procedure under test_level. Parameters excluded by coef_breaks are repeated across regime blocks. The optional spillover component is omitted when h = NULL. The fit has print(), summary(), plot() and the accessors in sysbreak-methods.

References

Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59(3), 817–858.

Andrews, D. W. K., and Monahan, J. C. (1992). An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica, 60(4), 953–966.

Zhu, X., Pan, R., Li, G., Liu, Y., and Wang, H. (2017). Network vector autoregression. The Annals of Statistics, 45(3), 1096–1123.

Qu, Z., and Perron, P. (2007). Estimating and testing structural changes in multivariate regressions. Econometrica, 75(2), 459–502.

Examples

set.seed(5)
bigt <- 80
W <- matrix(c(0, 1, 1, 0), 2, 2, byrow = TRUE)
y <- matrix(0, bigt, 2)
colnames(y) <- c("y1", "y2")
for (tt in 2:bigt) {
  alpha <- if (tt <= 40) 0.05 else 0.75
  beta <- 0.1
  y[tt, ] <- alpha * as.numeric(W %*% y[tt - 1, ]) +
    beta * y[tt - 1, ] + rnorm(2, sd = 0.2)
}

fit <- netbreak(
  y = y, W = W, lags = 1, m = 1,
  coef_breaks = c("network", "own"),
  trim = 0.2
)
fit$estimation$breaks

Simulated Network VAR

Description

One draw from Scenario 1 of Han and Lee (2024): a network VAR for 20 nodes over 300 periods, with breaks after periods 100 and 200.

Usage

netsim

Format

A list with two components:

y

A 300 by 20 matrix of observations, with columns node1 to node20.

W

The 20 by 20 row-normalized network matrix. Entry (i, l) is 1/n_i if node i follows node l and 0 otherwise, where n_i is the number of nodes that i follows.

Details

In regime j,

y_{it} = V_i'\gamma_j + \sum_{\ell} \alpha_{\ell,j} (W y_{t-\ell})_i + \sum_{\ell} \beta_{\ell,j} y_{i,t-\ell} + e_{it}, \qquad e_{it} \sim N(0, 0.1^2).

Regime 1 has one lag, with \alpha = -0.10 and \beta = 0.20. Regimes 2 and 3 have two lags, with \alpha = (0.20, -0.22), \beta = (-0.12, 0.40) and \alpha = (-0.12, 0.10), \beta = (0.25, -0.40). The covariates V_i are fixed, so V_i'\gamma_j is a node intercept that changes across regimes. The in-degrees of the network follow a power law, as in Zhu et al. (2017).

Source

Simulated with set.seed(1) following Han and Lee (2024).

References

Han, Y., and Lee, T. C. M. (2024). Structural break detection in non-stationary network vector autoregression models. IEEE Transactions on Network Science and Engineering, 11(5), 4134-4145.

Zhu, X., Pan, R., Li, G., Liu, Y., and Wang, H. (2017). Network vector autoregression. The Annals of Statistics, 45(3), 1096-1123.

Ordered Structural Breaks in a System of Equations

Description

Estimation of a multiple-equation system in which the coefficients of one group of equations change before those of another group. The two break dates are estimated jointly and are not restricted to coincide.

Usage

ordbreak(set1, set2, data,
  coef_breaks1 = "all", coef_breaks2 = "all",
  trim = 0.2, hetq = TRUE, max_break_gap = NULL,
  test_level = 0.05, ci_level = 0.95,
  confidence_intervals = FALSE,
  ci_grid_radius = 20, ci_replications = 5000, ci_step = 1000,
  ci_grid_step = 0.02, ci_seed = NULL, ci_verbose = FALSE,
  tol = 1e-8, max_iter = 1000)

Arguments

Details

The coefficients are stacked equation by equation in the order the formulas are given, so those of set1 occupy the leading q1 positions and those of set2 the remaining q2. ordcheck reports this layout without estimating anything, which is worth doing before a long run.

The test is that of Qu and Perron (2007, Section 6.4) for locally ordered breaks. The test for globally ordered breaks in their Section 6.5, whose limiting distribution differs, is not implemented.

Value

An object of class "ordbreak" containing:

spec

The setup: the sample size bigt, the number of equations n, the minimum segment length h, the number of coefficients in each set q1 and q2, the number of equations in the first set n1, the names of the coefficients allowed to change at each date, changing1 and changing2, the arguments trim, hetq and trun, the input formulas split into set1 and set2, the equation and coefficient names, the retained data rows y_rows after discarding missing values, and the selection matrix S.

testing

The SupLR statistic for ordered breaks, sup_lr_ordered, its critical_values at test_level, the pair truncated_breaks the statistic is computed at, and the log-likelihoods H1like and H0like it is built from. The critical value is indexed by the number of coefficients allowed to change, not by the total number of coefficients.

estimation

The estimated breaks, given as rows of data; the coefficients beta1 and beta2 of the two regimes; the residuals of the model at the estimated breaks, one column per equation; and the coefficient shifts delta1 and delta2 occurring at the two dates; and ci_breaks, a two by two matrix when confidence_intervals = TRUE and NULL otherwise, also given as rows of data. Its rows are the two break dates and its columns the lower and upper bounds at ci_level.

Note that the ordering is imposed rather than tested. The equations in set1 are taken to break first, so exchanging the two sets specifies a different model. Both specifications are tested against the same no-break null, and their values of sup_lr_ordered can be compared.

References

Qu, Z., and Perron, P. (2007). Estimating and testing structural changes in multivariate regressions. Econometrica, 75(2), 459–502.

Examples

set.seed(13)
bigt <- 60
x1 <- rnorm(bigt)
x2 <- rnorm(bigt)
y1 <- 1 + 0.5 * x1 + rnorm(bigt, sd = 0.3)
y2 <- 1 + 0.5 * x2 + rnorm(bigt, sd = 0.3)
y1[21:bigt] <- y1[21:bigt] + 1.5 * x1[21:bigt]
y2[36:bigt] <- y2[36:bigt] - 1.5 * x2[36:bigt]
dat <- data.frame(y1 = y1, y2 = y2, x1 = x1, x2 = x2)

fit <- ordbreak(y1 ~ x1, y2 ~ x2, data = dat, trim = 0.2)
fit$estimation$breaks

Coefficient Layout of an Ordered-Break Specification

Description

Reports the coefficients of an ordered-break model, which set of equations they belongs to, and which of them are allowed to change, without estimating anything.

Usage

ordcheck(set1, set2, data, coef_breaks1 = "all", coef_breaks2 = "all")

Arguments

Value

The layout is printed: one row per coefficient, giving its position in the coefficient vector, its name, the set it belongs to, whether it is allowed to change, and the date at which it changes.

Examples

set.seed(13)
dat <- data.frame(y1 = rnorm(40), y2 = rnorm(40),
                  x1 = rnorm(40), x2 = rnorm(40))

ordcheck(y1 ~ x1 + x2, y2 ~ x1 + x2, data = dat,
        coef_breaks1 = list(y1 ~ x1))

Plot a Structural-Break Fit

Description

Plots each series with the estimated break dates (dashed lines), their confidence intervals (shaded) and, optionally, the fitted values of the regime-specific model.

Usage

## S3 method for class 'sysbreak'
plot(x, which = NULL, dates = NULL, fitted = TRUE, ...)

Arguments

Value

x, invisibly.

See Also

sysbreak-methods, summary.sysbreak

Examples

set.seed(11)
bigt <- 40
x <- rnorm(bigt)
y1 <- 1 + 0.4 * x + rnorm(bigt, sd = 0.3)
y2 <- -0.5 + 0.2 * x + rnorm(bigt, sd = 0.3)
y1[21:40] <- y1[21:40] + 1.2 - 0.6 * x[21:40]
y2[21:40] <- y2[21:40] - 0.8 + 0.4 * x[21:40]
dat <- data.frame(y1 = y1, y2 = y2, x = x)
fit <- sysbreak(cbind(y1, y2) ~ x, dat, m = 1, trim = 0.3)
plot(fit)

Summarize a Structural-Break Fit

Description

A report of a fit returned by sysbreak, varbreak or netbreak: the break number, the break dates with their confidence intervals, and a coefficient table for each regime. print() of the fit gives the full output.

Usage

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

## S3 method for class 'summary.sysbreak'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

Arguments

Value

An object of class "summary.sysbreak", a list with the model type, the numbers of equations n and observations bigt, the tests for the break-number decision, the number of breaks selected, and, when at least one break is selected, the break dates with their confidence intervals (breaks), the rows of the data in each regime (regimes) and the coefficient tables (coefficients): estimate, standard error, z value and two-sided p-value, one table per regime, treating the break dates as known.

See Also

sysbreak-methods, plot.sysbreak

Examples

set.seed(11)
bigt <- 40
x <- rnorm(bigt)
y1 <- 1 + 0.4 * x + rnorm(bigt, sd = 0.3)
y2 <- -0.5 + 0.2 * x + rnorm(bigt, sd = 0.3)
y1[21:40] <- y1[21:40] + 1.2 - 0.6 * x[21:40]
y2[21:40] <- y2[21:40] - 0.8 + 0.4 * x[21:40]
dat <- data.frame(y1 = y1, y2 = y2, x = x)
fit <- sysbreak(cbind(y1, y2) ~ x, dat, m = 1, trim = 0.3)
summary(fit)

Structural Breaks in a System of Equations

Description

Estimation of a multiple-equation system affected by structural breaks in the regression coefficients and/or the variance covariance matrix of the residuals.

Usage

sysbreak(model, data = NULL, m, coef_breaks = "all",
  test_level = 0.05, ci_level = 0.95,
  trim = 0.15, verbose = FALSE, brv = FALSE,
  robust = c("none", "hc", "hac"), hetq = FALSE, prewhit = FALSE,
  restart = c("none", "brbp", "brbp_residual"),
  restart_B = 10, restart_verbose = FALSE)

Arguments

Details

The interpretation of testing$statistic depends on robust.

• robust = "none" gives the SupLR statistic. The code can test the coefficients specified by coef_breaks, the disturbance covariance when brv = TRUE, or both.

• robust = "hc" or "hac" gives the robust SupF statistic, for the coefficients only, because the covariance-break critical values are case dependent under heteroskedasticity or serial correlation; it assumes a constant error covariance. With brv = TRUE, the estimates, standard errors and confidence intervals account for the covariance breaks; under "hc" the intervals further require errors without ARCH-type dependence.

Value

An object of class "sysbreak" containing:

spec

A summary of the setup, including the sample size bigt, the number of equations n, the total number of regression coefficients per regime q, the arguments m, trim, brv, robust, hetq, prewhit, test_level and ci_level, the input formula or formulas, the equation and coefficient names, the retained data rows y_rows after discarding missing values, and a record of the coef_breaks selector.

testing

The tests, reported at test_level: statistic, testing no break against k breaks for every k = 1,\ldots,m, with its critical_values; statistic_type, "SupLR" or "SupF", and NA when a requested robust test is unavailable; wdmax and wdmax_critical_values; the sequential tests of l against l+1 breaks, each containing l, a statistic and its critical_values; and the number of changing_parameters, which includes the changing covariance parameters when robust = "none".

br_selected

The outcome of the break-number determination at test_level: the selected number of breaks, the WDmax statistic with its critical value and significance flag, and the sequential steps taken. breaks is 0 when WDmax is not significant, and NA when the tests are unavailable for the requested specification.

estimation

The estimates related to the selected number of breaks: estimated breaks; their confidence intervals ci_breaks; coefficients by regimes and their confidence intervals; disturbance covariance matrices. This component is NULL when no break is detected, in which case only testing and br_selected are informative.

y

The responses used in estimation, rows spec$y_rows of data.

Note that m is the maximum number of breaks allowed. Testing output is reported for every value from 1 through m, while estimation corresponds to the number of breaks selected by the procedure under test_level. The fit has print(), summary(), plot() and the accessors in sysbreak-methods.

References

Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59(3), 817–858.

Andrews, D. W. K., and Monahan, J. C. (1992). An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica, 60(4), 953–966.

Bai, J., and Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica, 66(1), 47–78.

Qu, Z., and Perron, P. (2007). Estimating and testing structural changes in multivariate regressions. Econometrica, 75(2), 459–502.

Examples

set.seed(11)
bigt <- 40
x <- rnorm(bigt)
y1 <- 1 + 0.4 * x + rnorm(bigt, sd = 0.3)
y2 <- -0.5 + 0.2 * x + rnorm(bigt, sd = 0.3)
y1[21:40] <- y1[21:40] + 1.2 - 0.6 * x[21:40]
y2[21:40] <- y2[21:40] - 0.8 + 0.4 * x[21:40]
dat <- data.frame(y1 = y1, y2 = y2, x = x)

fit <- sysbreak(cbind(y1, y2) ~ x, data = dat, m = 1, trim = 0.2)
fit$estimation$breaks

Methods for Displaying Structural-Break Fits

Description

Fits returned by sysbreak, varbreak and netbreak. All refer to the estimates at the selected number of breaks.

Usage

## S3 method for class 'sysbreak'
coef(object, ...)

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

## S3 method for class 'sysbreak'
confint(object, parm, level = NULL, type = c("breaks", "coef"), ...)

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

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

## S3 method for class 'sysbreak'
nobs(object, ...)

Arguments

Value

coef

A named vector of the regime coefficients: all coefficients of regime 1, then those of regime 2, and so on. Each name joins the regime and the coefficient label used by summary(), as in "regime1:y1:x" (equation y1, regressor x) or "regime2:mu:node3" for netbreak. Coefficients held constant by coef_breaks are repeated in every regime.

vcov

The covariance matrix of coef(), with rows and columns in the same order and with the same names. When coef_breaks holds coefficients constant, the matrix is singular, since the repeated entries of a constant coefficient are perfectly correlated.

confint

For type = "breaks", a matrix with one row per break: the lower bound, the estimated break date and the upper bound. For type = "coef", the lower and upper bounds of confidence intervals, estimate \pm normal quantile times standard error, treating the break dates as known.

residuals, fitted

Matrices with one column per equation and one row per observation; the row names are the rows of the data (spec$y_rows).

nobs

The number of observations used.

When no break is selected the regime estimates are not computed, and all but nobs stop with an error.

See Also

summary.sysbreak, plot.sysbreak

Examples

set.seed(11)
bigt <- 40
x <- rnorm(bigt)
y1 <- 1 + 0.4 * x + rnorm(bigt, sd = 0.3)
y2 <- -0.5 + 0.2 * x + rnorm(bigt, sd = 0.3)
y1[21:40] <- y1[21:40] + 1.2 - 0.6 * x[21:40]
y2[21:40] <- y2[21:40] - 0.8 + 0.4 * x[21:40]
dat <- data.frame(y1 = y1, y2 = y2, x = x)
fit <- sysbreak(cbind(y1, y2) ~ x, dat, m = 1, trim = 0.3)

coef(fit)
confint(fit)
confint(fit, level = 0.90)
confint(fit, type = "coef")
head(residuals(fit))
nobs(fit)

Structural Breaks in a VAR

Description

Estimation of a VAR affected by structural breaks in the regression coefficients and/or the variance covariance matrix of the residuals.

Usage

varbreak(y, lags, m, intercept = TRUE, coef_breaks = "all",
  test_level = 0.05, ci_level = 0.95,
  trim = 0.15, verbose = FALSE, brv = FALSE,
  robust = c("none", "hc", "hac"), hetq = FALSE, prewhit = FALSE,
  restart = c("none", "brbp", "brbp_residual"),
  restart_B = 10, restart_verbose = FALSE, h = NULL)

Arguments

Details

The interpretation of testing$statistic depends on robust.

• robust = "none" gives the SupLR statistic. The code can test the coefficients specified by coef_breaks, the disturbance covariance when brv = TRUE, or both.

• robust = "hc" or "hac" gives the robust SupF statistic, for the coefficients only, because the covariance-break critical values are case dependent under heteroskedasticity or serial correlation; it assumes a constant error covariance. With brv = TRUE, the estimates, standard errors and confidence intervals account for the covariance breaks; under "hc" the intervals further require errors without ARCH-type dependence.

Value

An object of class c("varbreak", "sysbreak") containing:

spec

A summary of the setup, including the sample size bigt, the number of equations n, the total number of coefficients per regime q, the arguments m, lags, intercept, trim, brv, robust, hetq, prewhit, test_level and ci_level, the equation and coefficient names, the retained input rows y_rows after discarding the lags, and the coef_breaks selector.

testing

The tests, reported at test_level: statistic, testing no break against k breaks for every k = 1,\ldots,m, with its critical_values; statistic_type, "SupLR" or "SupF", and NA when a requested robust test is unavailable; wdmax and wdmax_critical_values; the sequential tests of l against l+1 breaks, each containing l, a statistic and its critical_values; and the number of changing_parameters, which includes the changing covariance parameters when robust = "none".

br_selected

The outcome of the break-number determination at test_level: the selected number of breaks, the WDmax statistic with its critical value and significance flag, and the sequential steps taken. breaks is 0 when WDmax is not significant, and NA when the tests are unavailable for the requested specification.

estimation

The estimates related to the selected number of breaks: estimated breaks; their confidence intervals ci_breaks; coefficients by regimes and their confidence intervals; disturbance covariance matrices. This component is NULL when no break is detected, in which case only testing and br_selected are informative.

spillover

When h is supplied and at least one break is selected, regime-specific Diebold–Yilmaz generalized spillover measures returned by dyspill.

y

The responses used in estimation, rows spec$y_rows of y.

Note that m is the maximum number of breaks allowed. Testing output is reported for every value from 1 through m, while estimation corresponds to the number of breaks selected by the procedure under test_level. The optional spillover component is omitted when h = NULL. The fit has print(), summary(), plot() and the accessors in sysbreak-methods.

References

Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59(3), 817–858.

Andrews, D. W. K., and Monahan, J. C. (1992). An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica, 60(4), 953–966.

Bai, J., and Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica, 66(1), 47–78.

Qu, Z., and Perron, P. (2007). Estimating and testing structural changes in multivariate regressions. Econometrica, 75(2), 459–502.

Examples

set.seed(12)
bigt <- 45
y <- matrix(0, bigt, 2)
for (tt in 2:bigt) {
  shift <- if (tt > 24) c(0.8, -0.6) else c(0, 0)
  y[tt, ] <- shift +
    c(0.3 * y[tt - 1, 1], -0.2 * y[tt - 1, 2]) +
    rnorm(2, sd = 0.3)
}

fit <- varbreak(y, lags = 1, m = 1, trim = 0.2)
fit$estimation$breaks