mcusum_test function

Change Point Test for Regression

Apply change point test by if(!exists(".Rdpack.currefs")) .Rdpack.currefs <-new.env();Rdpack::insert_citeOnly(keys="Horvath_etal_2017;textual",package="funtimes",cached_env=.Rdpack.currefs)

for detecting at-most-$m$ changes in regression coefficients, where test statistic is a modified cumulative sum (CUSUM), and critical values are obtained with sieve bootstrap if(!exists(".Rdpack.currefs")) .Rdpack.currefs <-new.env();Rdpack::insert_citeOnly(keys="Lyubchich_etal_2020_changepoints",package="funtimes",cached_env=.Rdpack.currefs) .

mcusum_test(
  e,
  k,
  m = length(k),
  B = 1000,
  shortboot = FALSE,
  ksm = FALSE,
  ksm.arg = list(kernel = "gaussian", bw = "sj"),
  ...
)

Arguments

• e: vector of regression residuals (a stationary time series).

• k: an integer vector or scalar with hypothesized change point location(s) to test.

• m: an integer specifying the maximum number of change points being confirmed as statistically significant (from those specified in k) would be $\le m$. Thus, m must be in 1,...,k.

• B: number of bootstrap simulations to obtain empirical critical values. Default is 1000.

• shortboot: if TRUE, then a heuristic is used to perform the test with a reduced number of bootstrap replicates. Specifically, B/4 replicates are used, which may reduce computing time by up to 75% when the number of retained null hypotheses is large. A $p$-value of 999 is reported whenever a null hypothesis is retained as a result of this mechanism.

• ksm: logical value indicating whether a kernel smoothing to innovations in sieve bootstrap shall be applied (default is FALSE, that is, the original estimated innovations are bootstrapped, without the smoothing).

• ksm.arg: used only if ksm = TRUE. A list of arguments for kernel smoothing to be passed to density function. Default settings specify the use of the Gaussian kernel and the "sj" rule to choose the bandwidth.

• ...: additional arguments passed to ARest

  (for example, ar.method).

Returns

A list of class "htest" containing the following components: - method: name of the method.

• data.name: name of the data.

• statistic: obseved value of the test statistic.

• parameter: mhat is the final number of change points, from those specified in the input k, for which the test statistic is reported. See the corresponding locations, khat, in the estimate.

• p.value: bootstrapped $p$-value of the test.

• alternative: alternative hypothesis.

• estimate: list with elements: AR_order and AR_coefficients (the autoregressive order and estimated autoregressive coefficients used in sieve bootstrap procedure), khat (final change points, from those specified in the input k for which the test statistic is reported), and B (the number of bootstrap replications).

Details

The sieve bootstrap is applied by approximating regression residuals e

with an AR($p$) model using function ARest, where the autoregressive coefficients are estimated with ar.method, and order $p$ is selected based on ar.order and BIC settings (see ARest). At the next step, B autoregressive processes are simulated under the null hypothesis of no change points. The distribution of test statistics $M_T$ computed on each of those bootstrapped series is used to obtain bootstrap-based $p$-values for the test if(!exists(".Rdpack.currefs")) .Rdpack.currefs <-new.env();Rdpack::insert_citeOnly(keys="Lyubchich_etal_2020_changepoints",package="funtimes",cached_env=.Rdpack.currefs) .

In the current implementation, the bootstrapped $p$-value is calculated using equation 4.10 of if(!exists(".Rdpack.currefs")) .Rdpack.currefs <-new.env();Rdpack::insert_citeOnly(keys="Davison_Hinkley_1997;textual",package="funtimes",cached_env=.Rdpack.currefs) : p.value = (1 + $n$) / (B + 1), where $n$ is number of bootstrapped statistics greater or equal to the observed statistic.

The test statistic corresponds to the maximal value of the modified CUSUM over up to m combinations of hypothesized change points specified in k. The change points that correspond to that maximum are reported in estimate$khat, and their number is reported as the parameter.

Examples

##### Model 1 with normal errors, by Horvath et al. (2017)
T <- 100 #length of time series
X <- rnorm(T, mean = 1, sd = 1)
E <- rnorm(T, mean = 0, sd = 1)
SizeOfChange <- 1
TimeOfChange <- 50
Y <- c(1 * X[1:TimeOfChange] + E[1:TimeOfChange],
      (1 + SizeOfChange)*X[(TimeOfChange + 1):T] + E[(TimeOfChange + 1):T])
ehat <- lm(Y ~ X)$resid
mcusum_test(ehat, k = c(30, 50, 70))

#Same, but with bootstrapped innovations obtained from a kernel smoothed distribution:
mcusum_test(ehat, k = c(30, 50, 70), ksm = TRUE)

References

if(!exists(".Rdpack.currefs")) .Rdpack.currefs <-new.env();Rdpack::insert_all_ref(.Rdpack.currefs)

Author(s)

Vyacheslav Lyubchich