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Besides some regularity assumptions, white noise is a solution for a GARCH process. I have simulated a GARCH process that satisfies those regularity assumptions i.e. $\alpha + \beta < 1$. However, after applying the Priestley-Subba Rao (PSR) Test of Stationarity 10000 times(fractal package), I found that it rejects the assumption of stationarity 80% of the time (at a 5% significance level). My question is therefore:

What tests of stationarity can reject the assumption of stationarity 5% of the time at a 5% significance level, specifically for GARCH processes?

set.seed(1)

library(fractal)
library(fGarch)
library(tseries)

sp <- garchSpec(model = list(alpha = 0.5, beta = 0.4))

sig = 0.05
adf.garch <- c()
psr.garch <- c()


for(i in 1:1000){
  sm <- garchSim(spec = sp, n=1000)
  adf.garch[i] <- adf.test(sm$garch)$p.value < sig
  psr.garch[i] <- attr(stationarity(sm$garch), 'pvals')[1] < sig
}

sum(stat.garch) # stationary 100% of the time. Should be 95%.
1 - sum(psr.garch)/1000 # stationary 0.5% of the time. Should be 95%.
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