Automatic Bandwidth Selection

Description

Computes automatic bandwidth based on AR(1) approximation for each column of the input matrix. Uses Andrews (1991) method. Corresponds to GAUSS procedure bandw.

Usage

bandw(vhat)

Arguments

Value

Optimal bandwidth

Bootstrap Restart Algorithm

Description

Implements the Bootstrap Restarting to avoid local minima

Usage

brbp(
  y,
  z,
  q,
  m,
  bigt,
  trm,
  T_init,
  B = 20,
  S = NULL,
  s = NULL,
  R = NULL,
  r = NULL,
  verbose = FALSE
)

Arguments

Value

A list containing dx, delta, rssr, all_break_dates, and all_rssr.

References

Perron, P., and Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134(2), 373-399.

Residual Bootstrap Restarting Break-Point Algorithm

Description

Implements the Residual Bootstrap Restarting Break-Point algorithm to avoid local minima in restricted structural break estimation. Unlike the standard brbp algorithm, this method uses residual bootstrap instead of resampling observations, preserving the time ordering of the data.

Usage

brbp_residual(
  y,
  z,
  q,
  m,
  bigt,
  trm,
  T_init,
  B = 20,
  S = NULL,
  s = NULL,
  R = NULL,
  r = NULL,
  verbose = FALSE
)

Arguments

Value

A list containing dx, delta, rssr, all_break_dates, and all_rssr.

Build no-break (null) restrictions on \beta implied by R\delta = r

Description

Given linear restrictions on the break-model coefficient vector

R\delta = r, \qquad \delta \in \mathbb{R}^{(m+1)q},

this function constructs the implied restriction system under the no-break null, where all regimes share a common coefficient vector \beta\in\mathbb{R}^q.

Usage

build_null_from_R(R, r, m, q, tol)

Arguments

Details

Under no break, \delta = N\beta with N=\mathbf{1}_{m+1}\otimes I_q. Hence the implied null restrictions are

(RN)\beta = r.

The function reduces to independent rows and checks feasibility. It returns R_0\beta = r_0 with R_0 having q columns.

Value

A list with components:

• R0: Numeric matrix of dimension k_0 \times q giving the reduced null restrictions R_0\beta = r_0.

• r0: Numeric vector of length k_0.

• A_full: The unreduced matrix RN (dimension k\times q).

• rankA: Integer. Numerical rank of RN.

• p0: Integer. Number of free parameters under the null: p_0 = q - \mathrm{rank}(RN).

• restricted_null: Logical. TRUE iff p_0 < q.

Examples

m <- 1
q <- 2
R <- matrix(c(1, -1, 0, 0,
              0, 0, 1, -1), nrow = 2, byrow = TRUE)
r <- c(0, 0)
out <- build_null_from_R(R, r, m = m, q = q)
out$p0
out$R0
out$r0

Build no-break (null) restrictions implied by \delta = S\theta + s

Description

Given an affine restriction set for the break-model coefficients,

\delta = S\theta + s,

this function computes an affine parameterization for the no-break (null) model in terms of the common coefficient vector \beta\in\mathbb{R}^q:

\beta = S_0\gamma + s_0.

Usage

build_null_from_S(S, s, m, q, tol)

Arguments

Details

The null model is "no break" (all regimes share the same \beta) possibly with additional restrictions implied by S, s. The function returns S_0, s_0 and the number of free parameters under the null.

Under no break, the stacked coefficient vector satisfies

\delta = N\beta,\qquad N = \mathbf{1}_{m+1}\otimes I_q.

Combining with \delta = S\theta + s and eliminating \theta yields a linear system on \beta. This function constructs an affine parameterization \beta = S_0\gamma + s_0 for that system, or returns the unrestricted no-break model when no additional restrictions are implied.

The function stops with an error if the implied no-break system is infeasible.

Value

A list with components:

• S0: Numeric matrix of dimension q \times p_0 in \beta = S_0\gamma + s_0. If the null is unrestricted, S0 is diag(q).

• s0: Numeric vector of length q in \beta = S_0\gamma + s_0. If the null is unrestricted, s0 is a zero vector.

• p0: Integer. Number of free parameters under the null (equal to ncol(S0)).

• rA: Integer. Rank of the implied restriction system on \beta (so p_0 = q - r_A).

• restricted_null: Logical. TRUE if the null imposes restrictions beyond no-break (i.e., p0 < q), else FALSE.

Examples

m <- 1
q <- 2
S <- matrix(c(1, 0,
              1, 0,
              0, 1,
              0, 1), nrow = 4, byrow = TRUE)
s <- rep(0, 4)
out <- build_null_from_S(S, s, m = m, q = q)
out$p0
out$S0
out$s0

Compute Restricted Sum of Squared Residuals (RSSR)

Description

Computes the sum of squared residuals for given break dates and coefficients, taking into account restrictions if any.

Usage

compute_rssr(y, z, T_break, delta, m)

Compute HAC Variance-Covariance Matrix

Description

Main procedure for computing robust standard errors with optional AR(1) prewhitening. Corresponds to GAUSS procedure correct.

Usage

correct(reg, res, prewhit)

Arguments

Value

HAC variance-covariance matrix (d x d)

Critical Values for Break Date Confidence Intervals

Description

Computes critical values for confidence intervals of break dates. Corresponds to GAUSS procedure cvg.

Usage

cvg(eta, phi1s, phi2s)

Arguments

Value

Vector of critical values (4 x 1) for 2.5%, 5%, 95%, 97.5% quantiles

Find Optimal Break Dates (Unrestricted)

Description

Finds break dates that globally minimize the sum of squared residuals (SSR) without any restrictions on coefficients, using dynamic programming.

Usage

dating(y, z, h, m, q, bigt)

Arguments

Details

This function implements the dynamic programming algorithm of Bai and Perron (1998, 2003) to efficiently find the optimal break dates. The algorithm:

Value

A list containing:

References

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

Bai, J., and Perron, P. (2003). Computation and analysis of multiple structural change models. Journal of Applied Econometrics, 18(1), 1-22.

Find Optimal Break Dates (Under Restrictions)

Description

Finds break dates that minimize the SSR given pre-estimated restricted coefficients, using dynamic programming.

Usage

dating2(bigvec2, h, m, bigt)

Arguments

Value

A list containing:

References

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

Perron, P., and Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134(2), 373-399.

Taylor Rule Deviation Data

Description

A dataset containing quarterly Taylor-rule deviations used in the restricted structural break illustration.

Usage

deviation

Format

A data frame with 193 rows and 2 variables:

date

Quarterly date labels.

y

Taylor-rule deviation series used in the application example.

Source

Constructed from the replication files for Nikolsko-Rzhevskyy et al. (2014).

References

Nikolsko-Rzhevskyy, A., Papell, D. H., and Prodan, R. (2014). Deviations from rules-based policy and their effects. Journal of Economic Dynamics and Control, 49, 4–17.

Examples

data(deviation)
head(deviation)

Estimate Breaks and Coefficients (Form A)

Description

Estimates break dates and coefficients under Form A restrictions \delta = S \theta + s without computing standard errors or confidence intervals.

Usage

est(y, z, q, m, bigt, trm, S, s, T_init = NULL)

Arguments

Value

A list containing:

• dx: A numeric vector (m \times 1) of estimated break dates

• delta: A numeric vector ((m+1)q times 1) of estimated coefficients under restrictions. Coefficients are organized by regime: first q elements for regime 1, next q for regime 2, etc.

References

Perron, P., and Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134(2), 373-399.

Estimate Breaks and Coefficients (Form B)

Description

Estimates break dates and coefficients under Form B restrictions R \delta = r without computing standard errors or confidence intervals.

Usage

est2(y, z, q, m, bigt, trm, R, r, T_init = NULL)

Arguments

Value

A list containing:

• dx: Estimated break dates (m x 1 vector). The i-th element gives the observation number where the i-th break occurs.

• delta: Estimated coefficients under restrictions ((m+1)*q x 1 vector). Coefficients are organized by regime: first q elements for regime 1, next q for regime 2, etc.

References

Perron, P., and Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134(2), 373-399.

See Also

estco2 for estimation with standard errors and confidence intervals

Estimation with Standard Errors and Confidence Intervals (Form A)

Description

Estimates break dates and coefficients under Form A restrictions, computes HAC standard errors, and constructs confidence intervals for break dates.

Usage

estco(m, q, z, y, trm, robust, prewhit, hetvar, S, s, Verbose)

Arguments

Value

A list (returned invisibly) with the following components:

breaks

An integer vector of length m giving the estimated break dates.

coefficients

A numeric vector of length (m+1)q giving the regime-specific coefficient estimates.

vcov

A (m+1)q \times (m+1)q variance-covariance matrix of coefficients.

ci_breaks

An m \times 4 matrix of confidence intervals for the break dates. The first two columns give the 95\ (lower bound, upper bound) and the last two columns give the 90\ (lower bound, upper bound).

References

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

Perron, P., and Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134(2), 373-399.

Examples

data(example)
y <- example$y
z <- matrix(1, length(y), 1)
m <- 3
q <- 1
trm <- 0.10
S <- matrix(c(1, 0,
              0, 1,
              1, 0,
              0, 1), nrow = 4, byrow = TRUE)
s <- rep(0, 4)
fit <- estco(m, q, z, y, trm, 0, 0, 1, S, s, FALSE)
fit$breaks

Estimation with Standard Errors and Confidence Intervals (Form B)

Description

Estimates break dates and coefficients under Form B restrictions, computes HAC standard errors, and constructs confidence intervals for break dates.

Usage

estco2(m, q, z, y, trm, robust, prewhit, hetvar, R, r, Verbose)

Arguments

Value

A list (returned invisibly) with the following components:

breaks

An integer vector of length m giving the estimated break dates.

coefficients

A numeric vector of length (m+1)q giving the regime-specific coefficient estimates.

vcov

A (m+1)q \times (m+1)q variance-covariance matrix of coefficients.

ci_breaks

An m \times 4 matrix of confidence intervals for the break dates. The first two columns give the 95\ (lower bound, upper bound) and the last two columns give the 90\ (lower bound, upper bound).

References

Perron, P., and Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134(2), 373-399.

Examples

data(example)
y <- example$y
z <- matrix(1, length(y), 1)
m <- 3
q <- 1
trm <- 0.10
R <- matrix(c(1, 0, -1, 0,
              0, 1,  0, -1), nrow = 2, byrow = TRUE)
r <- c(0, 0)
fit <- estco2(m, q, z, y, trm, 0, 0, 1, R, r, FALSE)
fit$breaks

Estimation with Standard Errors and Confidence Intervals (Form B)

Description

Estimates break dates and coefficients under Form B restrictions, computes HAC standard errors, and constructs confidence intervals for break dates.

Usage

estco2_fb(
  m,
  q,
  z,
  y,
  trm,
  T_init,
  robust,
  prewhit,
  hetvar,
  R,
  r,
  Verbose = FALSE
)

Arguments

Value

A list (returned invisibly) containing breaks, coefficients, vcov, and ci_breaks.

References

Perron, P., and Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134(2), 373-399.

Estimation with Standard Errors and Confidence Intervals (Form A), conditional on estimated breaks

Description

Estimates coefficients under Form A restrictions, computes HAC standard errors, and constructs confidence intervals for break dates.

Usage

estco_fb(
  m,
  q,
  z,
  y,
  trm,
  T_init,
  robust,
  prewhit,
  hetvar,
  S,
  s,
  Verbose = FALSE
)

Arguments

Value

A list (returned invisibly) containing breaks, coefficients, vcov, and ci_breaks.

References

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

Perron, P., and Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134(2), 373-399.

Example Time Series Data

Description

A dataset containing 120 observations of a time series variable for illustrating restricted structural change models.

Usage

example

Format

A data frame with 120 rows and 1 variable:

y

Numeric vector containing the time series observations

Details

This dataset is used in the examples in Perron and Qu (2006).

Source

Included with the rbreak package for illustration purposes.

References

Perron, P., and Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134(2), 373-399.

Examples

# Load the example data
data(example)


# Example: Test for 3 breaks with intercept only model
y <- example$y
z <- matrix(1, length(y), 1)  # Intercept only

# Form B restrictions: coefficients in regimes 1 and 3 are equal
m <- 3
q <- 1
R <- matrix(c(1, 0, -1, 0, 0, 1, 0, -1), nrow = 2, byrow = TRUE)
r <- c(0, 0)

result <- mainp(m = m, q = q, z = z, y = y, trm = 0.10,
                robust = 1, prewhit = 0, hetvar = 1,
                R = R, r = r,
                doestim = 1, dotest = 1, docv = 0,
                formb = 1)

Helper Function for Critical Value Calculation

Description

Evaluates the distribution function for computing critical values. Corresponds to GAUSS procedure funcg.

Usage

funcg(x, bet, alph, b, deld, gam)

Arguments

Value

Function value

Confidence Intervals for Break Dates

Description

Computes confidence intervals for estimated break dates using the shrinking shifts asymptotic framework.

Usage

interval(y, z, zbar, b, q, m, delta, robust, prewhit, hetomega)

Arguments

Value

A numeric matrix (m \times 4) with confidence interval bounds for each break date.

References

Bai, J. (1997). Estimation of a change point in multiple regression models. Review of Economics and Statistics, 79(4), 551-563.

Perron, P., and Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134(2), 373-399.

Long-Run Covariance Matrix (Newey-West style)

Description

Computes the long-run covariance matrix using quadratic spectral kernel with automatic bandwidth selection. Corresponds to GAUSS procedure jhatpr.

Usage

jhatpr(vmat)

Arguments

Value

Long-run covariance matrix (d x d)

Quadratic Spectral Kernel

Description

Evaluates the quadratic spectral kernel at value x. Corresponds to GAUSS procedure kern.

Usage

kern(x)

Arguments

Value

Kernel weight

Linear Regression Tree

Description

Constructs a regression tree where each terminal node fits a linear model. The tree is built by greedy recursive binary partitioning: at each node, every partition variable is considered as a candidate split axis. For numeric variables, data are sorted by the candidate variable and an optimal one-break point is found (minimising SSR via ssr()). For categorical variables, all possible binary partitions of the levels are enumerated. BIC is used to decide whether splitting improves the fit. The process continues until no further BIC improvement is possible.

Usage

ltree(y, z, partition_vars, cat_vars, trm, max_depth, min_obs,
  prune, verbose)

Arguments

Details

Each node in the tree is a list with type = "leaf" or type = "internal".

A leaf node contains: n, indices, coefficients, ssr, bic.

An internal node contains: split_var, split_var_name, split_type ("numeric" or "categorical"), split_val (for numeric) or split_left_levels (for categorical), bic_nosplit, bic_split, left, right, n.

The BIC criterion used is:

\mathrm{BIC} = n \log(\mathrm{SSR}/n) + k \log(n),

where k = q for no split and k = 2q for a single split.

Value

An object of class "ltree", a list containing:

tree

The recursive tree structure (see Details).

n

Total number of observations.

q

Number of regressors.

p

Number of partition variables.

pv_names

Names of the partition variables.

z_names

Names of the regressors.

is_cat

Logical vector indicating which partition variables are categorical.

trm

Trimming parameter used.

max_depth

Maximum depth used.

min_obs

Minimum observations per segment used.

call

The matched call.

S3 Methods

print(x, ...)

Prints a summary of the fitted tree, including the number of leaves, tree depth, and the split structure.

predict(object, newz, new_partition_vars = NULL, ...)

Returns predicted values for new data. newz is a numeric matrix of regressors and new_partition_vars is an optional data frame of partition variables (defaults to newz).

plot(x, data = x$partition_vars, cex = 0.85, main = "", ...)

Draws a tree diagram of the fitted segmented linear regression tree. Internal nodes are shown as beige rectangles, leaf nodes as green ovals. data defaults to the partition variables stored in the fitted object, so plot(res) works out of the box. Right-side levels of categorical splits are labelled automatically.

References

Perron, P., and Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134(2), 373-399.

Quinlan, J.R. (1992). Learning with continuous classes. In: Proceedings of the Australian Joint Conference on Artificial Intelligence, World Scientific, pp 343–348.

Examples

data(ltree1)
fit <- ltree(ltree1$y[1:120],
             ltree1[1:120, c("x1", "x2", "x3", "x4")],
             trm = 0.1, max_depth = 2, prune = TRUE)
print(fit)

Linear Model Tree Data

Description

A dataset containing one realization from the piecewise linear design used to illustrate the generalized regression tree (linear model tree).

Usage

ltree1

Format

A data frame with 1500 rows and 5 variables:

y

Response variable.

x1

Continuous regressor.

x2

Continuous regressor.

x3

Continuous regressor.

x4

Categorical regressor.

Details

The data are generated from a segmented linear regression function with true splits at X_1 = 10, X_2 = 10, X_2 = 15, and X_4 \in \{a,b\} versus \{c\}.

Source

Generated from the design in Zheng and Chen (2019) and included for illustration purposes.

References

Quinlan, J.R. (1992). Learning with continuous classes. In: Proceedings of the Australian Joint Conference on Artificial Intelligence, World Scientific, pp 343–348.

Zheng, X. and Chen, S. X. (2019). Partitioning structure learning for segmented linear regression trees. Advances in Neural Information Processing Systems (NeurIPS 2019).

Examples

data(ltree1)
fit <- ltree(ltree1$y, ltree1[, c("x1", "x2", "x3", "x4")],
             trm = 0.1, max_depth = 10, prune = TRUE)
print(fit)

Smooth Surface Regression Tree Example Data

Description

A dataset containing one fixed sample for the smooth-surface generalized regression tree illustration.

Usage

ltree2

Format

A data frame with 1000 rows and 3 variables:

y

Response variable with Gaussian noise.

x1

Continuous regressor.

x2

Continuous regressor.

Details

The underlying regression surface is

m(X) = \max\{e^{-10X_1^2}, e^{-50X_2^2}, 1.25e^{-5(X_1^2 + X_2^2)}\}.

Source

Generated from the design in Zheng and Chen (2019) and included for illustration purposes.

References

Quinlan, J.R. (1992). Learning with continuous classes. In: Proceedings of the Australian Joint Conference on Artificial Intelligence, World Scientific, pp 343–348.

Zheng, X. and Chen, S. X. (2019). Partitioning structure learning for segmented linear regression trees. Advances in Neural Information Processing Systems (NeurIPS 2019).

Examples

data(ltree2)
fit <- ltree(ltree2$y,
             cbind(1, ltree2$x1, ltree2$x2),
             partition_vars = ltree2[, c("x1", "x2")],
             trm = 0.1, max_depth = 10, min_obs = 10, prune = TRUE)
print(fit)

Main Procedure for Restricted Structural Change Analysis

Description

Function for restricted structural change models that performs estimation, hypothesis testing, and critical value simulation.

Usage

mainp(m, q, z, y, trm, robust, prewhit, hetvar, S, s, 
  R, r, doestim, dotest, docv, Tstar, rep, bigt,
  forma, formb, Verbose)

Arguments

Details

The function supports two forms of linear restrictions on the break-model coefficient vector \delta \in \mathbb{R}^{(m+1)q}:

• Form A: \delta = S\theta + s, an affine parameterization with basis matrix S and shift s. Use forma = 1.

• Form B: R\delta = r, explicit linear constraints. Use formb = 1.

Both forms can be used simultaneously (forma = 1 and formb = 1), in which case both sets of results are computed and the last one overwrites shared list entries (estimation, test_statistic, critical_values). If both forms are used simultaneously, consider storing results separately.

For testing and critical value simulation, the implied no-break null restrictions on the common coefficient vector \beta are derived automatically via build_null_from_S() (Form A) or build_null_from_R() (Form B).

Value

A list (returned invisibly) with some or all of the following components, depending on the values of doestim, dotest, docv, forma, and formb:

• estimation: Output from the estimation procedure. Produced when doestim = 1. If forma = 1, this is the output of estco(); if formb = 1, this is the output of estco2(). A list containing:

  • breaks: An integer vector of length m giving the estimated break dates.

  • coefficients: A numeric vector of length (m+1)q giving the regime-specific coefficient estimates.

  • vcov: A (m+1)q \times (m+1)q variance-covariance matrix of coefficients.

  • ci_breaks: An m \times 4 matrix of confidence intervals for the break dates. The first two columns give the 95\ (lower bound, upper bound) and the last two columns give the 90\ (lower bound, upper bound).

• test_statistic: The value of the restricted structural change sup-F test statistic. A scalar. Produced when dotest = 1. If forma = 1, this is the output of pftest(); if formb = 1, this is the output of pftest2(). The null hypothesis is no structural break subject to the implied null restrictions derived from S, s (Form A) or R, r (Form B) via build_null_from_S() or build_null_from_R().

• critical_values: A numeric vector of length 4 containing simulated critical values of the sup-F test at the 90\ in that order. Produced when docv = 1. If forma = 1, simulated by rcv(); if formb = 1, simulated by rcv2(). Obtained by Monte Carlo simulation with rep replications and sample size Tstar, under the null of no break with the implied null restrictions.

References

Perron, P., and Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134(2), 373-399.

Examples

data(example)
y <- example$y
z <- matrix(1, length(y), 1)
m <- 3
q <- 1
trm <- 0.10
R <- matrix(c(1, 0, -1, 0,
              0, 1,  0, -1), nrow = 2, byrow = TRUE)
r <- c(0, 0)
out <- mainp(m = m, q = q, z = z, y = y, trm = trm,
             robust = 0, prewhit = 0, hetvar = 1,
             R = R, r = r,
             doestim = 1, dotest = 0, docv = 0,
             bigt = length(y), forma = 0, formb = 1, Verbose = FALSE)
out$estimation$breaks

Map Bootstrap Break Dates to Original Time Scale

Description

Maps break dates from bootstrap sample back to original time indices. This is the key function: if break point is at position k in bootstrap sample (which has time indices sorted as orig_time_map), then the break date in original time scale is orig_time_map[k].

Usage

map_boot_break_dates_to_original(T_boot, orig_time_map, bigt, h)

Map Break Dates to Bootstrap Sample

Description

Maps break dates from original sample to bootstrap sample indices.

Usage

map_break_dates_to_boot_sample(T_orig, orig_time_map, bigt, h)

Bootstrap restart optimization

Description

Applies bootstrap-based restart and re-optimization after an initial run of mainp. The procedure can be applied taking the break estimates from mainp as starting values. It generates bootstrap samples, re-estimates the break dates, and plugs the new estimates back into the original sample as starting values for re-optimization. This process is repeated to reduce the chance of convergence to a spurious local optimum.

Usage

mbrbp(m, q, z, y, trm, B, T_init, robust, prewhit, hetvar,
  S, s, R, r, doestim, dotest, docv, cvs,
  Tstar, rep, bigt, forma, formb, verbose, resi)

Arguments

Value

(Invisibly) a list with some or all of the following components, depending on the values of doestim, dotest, docv, forma, and formb:

• estimation: Output from the estimation procedure. Produced when doestim = 1. If forma = 1, this is the output of estco(); if formb = 1, this is the output of estco2(). A list containing:

  • breaks: An integer vector of length m giving the re-optimized break date estimates.

  • coefficients: A numeric vector of length (m+1)q giving the regime-specific coefficient estimates.

  • vcov: A (m+1)q \times (m+1)q variance-covariance matrix of coefficients.

  • ci_breaks: An m \times 4 matrix of confidence intervals for the break dates. The first two columns give the 95\ (lower bound, upper bound) and the last two columns give the 90\ (lower bound, upper bound).

• test_statistic: The value of the restricted structural change sup-F test statistic. A scalar. Produced when dotest = 1. If forma = 1, this is the output of pftest(); if formb = 1, this is the output of pftest2(). The null hypothesis is no structural break subject to the implied null restrictions derived from S, s (Form A) or R, r (Form B) via build_null_from_S() or build_null_from_R().

• critical_values: A numeric vector of length 4 containing critical values of the sup-F test at the 90\ that order. Produced when docv = 1. If cvs is supplied, these are returned directly without simulation; otherwise obtained by Monte Carlo simulation with rep replications and sample size Tstar, under the null of no break with the implied null restrictions. If forma = 1, simulated by rcv(); if formb = 1, simulated by rcv2().

References

Perron, P., and Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134(2), 373–399.

Wood, S. N. (2001). Minimizing model fitting objectives that contain spurious local minima by bootstrap restarting. Biometrics, 57(1), 240–244.

Examples

data(example)
y <- example$y
z <- matrix(1, length(y), 1)
m <- 3
q <- 1
trm <- 0.10
R <- matrix(c(1, 0, -1, 0,
              0, 1,  0, -1), nrow = 2, byrow = TRUE)
r <- c(0, 0)
init <- mainp(m = m, q = q, z = z, y = y, trm = trm,
              robust = 0, prewhit = 0, hetvar = 1,
              R = R, r = r,
              doestim = 1, dotest = 0, docv = 0,
              bigt = length(y), forma = 0, formb = 1, Verbose = FALSE)
out <- mbrbp(m = m, q = q, z = z, y = y, trm = trm, B = 3,
             T_init = init$estimation$breaks,
             robust = 0, prewhit = 0, hetvar = 1,
             R = R, r = r,
             doestim = 1, dotest = 0, docv = 0,
             bigt = length(y), forma = 0, formb = 1, verbose = FALSE)
out$estimation$breaks

Simple OLS Regression

Description

Performs ordinary least squares regression. Corresponds to the GAUSS procedure olsqr.

Usage

olsqr(y, x)

Arguments

Value

OLS coefficient estimates (k x 1)

OLS Regression with Residuals and Fitted Values

Description

Performs OLS regression and returns coefficients, residuals, and fitted values. Corresponds to the GAUSS procedure olsqr2. Uses pseudoinverse if matrix is singular (like GAUSS invpd).

Usage

olsqr2(y, x)

Arguments

Value

List with:

Optimal One-Break Partition (Unrestricted)

Description

Obtains an optimal one-break partition for a segment that starts at date start and ends at date last. Corresponds to GAUSS procedure parti.

Usage

parti(start, b1, b2, last, bigvec, bigt)

Arguments

Value

List with:

Optimal One-Break Partition (Restricted)

Description

Obtains an optimal one-break partition for a segment under given coefficients. Called only when number of breaks is one. Corresponds to GAUSS procedure parti2.

Usage

parti2(start, b1, b2, last, bigvec2, bigt)

Arguments

Value

List with:

Restricted Structural Change Sup-F Test (Form A)

Description

Constructs the supremum F-test for structural change under Form A restrictions.

Usage

pftest(S, S0, y, z, m, q, bigt, trm, robust, prewhit, hetvar, s, s0,
  Verbose)

Arguments

Value

The F test statistic (returned invisibly).

References

Perron, P., and Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134(2), 373-399.

Examples

data(example)
y <- example$y
z <- matrix(1, length(y), 1)
m <- 3
q <- 1
trm <- 0.10
S <- matrix(c(1, 0,
              0, 1,
              1, 0,
              0, 1), nrow = 4, byrow = TRUE)
s <- rep(0, 4)
null_model <- build_null_from_S(S, s, m, q)
pftest(S, null_model$S0, y, z, m, q, length(y), trm,
       0, 0, 1, s, null_model$s0, FALSE)

Restricted Structural Change Sup-F Test (Form B)

Description

Constructs the supremum F-test for structural change under Form B restrictions.

Usage

pftest2(R, R0, y, z, m, q, bigt, trm, robust, prewhit, hetvar, r, r0,
  Verbose)

Arguments

Value

The F test statistic (returned invisibly).

References

Perron, P., and Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134(2), 373-399.

Examples

data(example)
y <- example$y
z <- matrix(1, length(y), 1)
m <- 3
q <- 1
trm <- 0.10
R <- matrix(c(1, 0, -1, 0,
              0, 1,  0, -1), nrow = 2, byrow = TRUE)
r <- c(0, 0)
null_model <- build_null_from_R(R, r, m, q)
pftest2(R, null_model$R0, y, z, m, q, length(y), trm,
        0, 0, 1, r, null_model$r0, FALSE)

Compute F-statistic given break dates

Description

Compute F-statistic given break dates

Usage

pftest_fb(
  y,
  z,
  m,
  q,
  bigt,
  trm,
  S,
  s,
  S0,
  T_init,
  robust = 1,
  prewhit = 0,
  hetvar = 1,
  Verbose = FALSE
)

Arguments

Value

F-test statistic value

Pseudo-Inverse

Description

Computes the pseudo-inverse (Moore-Penrose inverse) of a matrix. Corresponds to GAUSS function pinv.

Usage

pinv(A)

Arguments

Value

Pseudo-inverse matrix

Diagonal Matrix of Segment Proportions

Description

Constructs a diagonal matrix with (T_i - T_(i-1))/T on the diagonal. Corresponds to GAUSS procedure plambda.

Usage

plambda(b, m, bigt)

Arguments

Value

Diagonal matrix (m+1 x m+1)

Diagonal Matrix of Segment-Specific Residual Variances

Description

Computes a diagonal matrix with the variance of residuals for each segment. Corresponds to GAUSS procedure psigmq.

Usage

psigmq(res, b, q, m, nt)

Arguments

Value

Diagonal matrix (m+1 x m+1)

Variance-Covariance Matrix of Delta

Description

Computes the variance-covariance matrix of the coefficient estimates. Corresponds to GAUSS procedure pvdel.

Usage

pvdel(y, z, i, q, bigt, b, coeff, prewhit, robust, hetvar)

Arguments

Value

Variance-covariance matrix ((m+1)*q x (m+1)*q)

Create diagonal partition of z matrix

Description

Constructs the diagonal block matrix Z_bar for m breaks. Corresponds to the GAUSS procedure pzbar.

Usage

pzbar(zz, m, bb)

Arguments

Value

Diagonal block matrix Z_bar (T x (m+1)*q)

Qu's Algorithm Wrapper

Description

Wrapper function for Qu's algorithm (est or est2) with initial break dates.

Usage

qu_algorithm(y, z, q, m, bigt, trm, T_init, S, s, R, r)

Simulate Critical Values of Sup-F Test (Form A)

Description

Simulates critical values for the restricted structural change sup-F test using Monte Carlo methods with Form A restrictions.

Usage

rcv(Tstar, rep, q, m, S, s, S0, s0, trm, Verbose)

Arguments

Value

A numeric vector of length 4 containing critical values at 90%, 95%, 97.5%, and 99% quantiles.

References

Perron, P., and Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134(2), 373-399.

Examples

m <- 3
q <- 1
trm <- 0.10
S <- matrix(c(1, 0,
              0, 1,
              1, 0,
              0, 1), nrow = 4, byrow = TRUE)
s <- rep(0, 4)
null_model <- build_null_from_S(S, s, m, q)
rcv(Tstar = 40, rep = 5, q = q, m = m,
    S = S, s = s, S0 = null_model$S0, s0 = null_model$s0,
    trm = trm, Verbose = FALSE)

Simulate Critical Values of Sup-F Test (Form B)

Description

Simulates critical values for the restricted structural change sup-F test using Monte Carlo methods with Form B restrictions.

Usage

rcv2(Tstar, rep, q, m, R, r, R0, r0, trm, Verbose)

Arguments

Value

A numeric vector of length 4 containing critical values at 90%, 95%, 97.5%, and 99% quantiles.

References

Perron, P., and Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134(2), 373-399.

Examples

m <- 3
q <- 1
trm <- 0.10
R <- matrix(c(1, 0, -1, 0,
              0, 1,  0, -1), nrow = 2, byrow = TRUE)
r <- c(0, 0)
null_model <- build_null_from_R(R, r, m, q)
rcv2(Tstar = 40, rep = 5, q = q, m = m,
     R = R, r = r, R0 = null_model$R0, r0 = null_model$r0,
     trm = trm, Verbose = FALSE)

Simulate Critical Values of Sup-F Test (Form B) - Parallel Version

Description

Simulates critical values using parallel processing for faster computation.

Usage

rcv2_parallel(Tstar, rep, q, m, R, r, R0, r0, trm, n_cores = NULL)

Arguments

Value

A numeric vector of length 4 containing critical values at 90%, 95%, 97.5%, and 99% quantiles.

Simulate Critical Values of Sup-F Test (Form A) - Parallel Version

Description

Simulates critical values using parallel processing for faster computation.

Usage

rcv_parallel(Tstar, rep, q, m, S, s, S0, s0, trm, n_cores = NULL)

Arguments

Value

A numeric vector of length 4 containing critical values at 90%, 95%, 97.5%, and 99% quantiles.

Recursive Sum of Squared Residuals

Description

Computes recursive residuals from a data set starting at date "start" and ending at date "last".

Usage

ssr(start, y, z, h, last)

Arguments

Value

Vector of SSR values (last x 1). First h-1 elements have no meaning.

SSR for All Possible Segments

Description

Creates a matrix storing the SSR for all possible segments under estimated coefficients. Corresponds to GAUSS procedure ssr2.

Usage

ssr2(y, z, b, q, m)

Arguments

Value

Vector of SSR values (T*(m+1) x 1)

Utility Functions for Restricted Structural Change Models

Description

Utility Functions for Restricted Structural Change Models