# LBI_test: Locally best invariant test against a change in persistence In memochange: Testing for Structural Breaks under Long Memory and Testing for Changes in Persistence

## Description

This function performs the locally best invariant test against a change in persistence as suggested by Busetti and Taylor (2004). Under the null hypothesis the time series is I(0) throughout and under the alternative a change from either I(1) to I(0) or I(0) to I(1) has occured.

## Usage

 ```1 2``` ```LBI_test(x, trend = c("none", "linear"), tau = 0.2, statistic = c("mean", "max", "exp"), simu = 0, M = 10000) ```

## Arguments

 `x` the univariate numeric vector to be investigated. Missing values are not allowed. `trend` whether the time series exhibits a trend, `"none"` implies no trend and `"linear"` implies a linear trend. `tau` the function tests in the interval `[T*tau,T*(1-tau)]` for a break in persistence with T being the length of the time series. It must hold that `0

## Details

The critical values of the tests vary with the sample size. If `simu=0`, the critical values provided are based on linear interpolation of the critical values simulated by Busetti and Taylor (2004). These are, however, only valid for `tau=0.2`. In case that another value is chosen for `tau`, it is recommended to set `simu=1` which means that critical values are simulated based on the given data using M replications. For a time series of length `T=100` and `M=10,000` replications this takes approximately five minutes with increasing duration for higher T or M. It should be noted, however, that M smaller than 10,000 make the results unreliable.

## Value

Returns a matrix that consists of test statistic and critical values (corresponding to `alpha=0.1,0.05,0.01`) for testing against a change from I(1) to I(0), I(0) to I(1), and against a change in an unknown direction.

Janis Becker

## References

Busetti, F. and Taylor, R. (2004): Tests of stationarity against a change in persistence. Journal of Econometrics, 123, pp. 33-66.

`cusum_test`, `LKSN_test`, `MR_test`, `ratio_test`.
 ```1 2 3 4 5 6 7``` ```set.seed(410) # generate dummy-data series <- c(rnorm(100), cumsum(rnorm(100))) # test for a break in persistence LBI_test(series, trend="none", statistic="mean") ```