# Why are forecasts from ARIMA and ETS equivalents different?

ETS models have ARIMA equivalents - this is described, eg, here and here.

However when fitting pairs of ARIMA and ETS equivalents in R I sometimes get different results. For example, compare forecasts from ARIMA(0,2,2) and ETS(AAN):

#Simulate random walk
n <- 100
x <- cumsum(rnorm(n))

#Fit models
arimafit <- Arima(x, order=c(0,2,2), include.constant = FALSE)
etsfit <- ets(x ,"AAN", damped = FALSE)

#Plot results
plot(x, type = "l", xlim = c(0, n + (n/5)))
lines(c(rep(NA, n), forecast(arimafit, h = n/5)$mean), col = "red") lines(c(rep(NA, n), forecast(etstrend, h = n/5)$mean), col = "blue")


I understand that this equivalence is conditional - ie the models can yield identical results given certain coefficient values. I am looking to understand what conditions result in more similar ETS/ARIMA forecasts and what conditions lead to divergence.