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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.

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Not all ETS models have ARIMA equivalents. Read the first link you posted more carefully. Also, the parameterization of the equivalent models is different, and the optimization is different.So you would not expect the forecasts to be identical, even when the models are algebraically equivalent.

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