Hi,

I have simulated data as follows:

```
#===============================
# In base R
# simulate trend-stationary or deterministic trend process
set.seed(47523)
sigma <- 3.0
nburn <- 100
nobs <- 400
e <- rnorm(n = 500, mean = 0, sd = sigma)
trend <- c(1:(nburn+nobs))
# constant + deterministic trend + white noise
tS <- as.ts(1.0 + 0.4*trend + e)
tS
plot.ts(tS[(nburn+1):(nburn+nobs)])
```

Based on fpp3 10.4 Stochastic and deterministic trends | Forecasting: Principles and Practice (3rd ed) I have modelled the data as

```
#===============================
library(fable)
fit_deterministic <- tS |>
as_tsibble() |>
tail(400) |>
model(deterministic = ARIMA(value ~ 1 + trend() + pdq(d = 0)))
report(fit_deterministic)
fit_stochastic <- tS |>
as_tsibble() |>
tail(400) |>
model(stochastic = ARIMA(value ~ pdq(d = 1)))
report(fit_stochastic)
```

with results

```
Series: value
Model: LM w/ ARIMA(0,0,2) errors
Coefficients:
ma1 ma2 trend() intercept
-0.0947 -0.1506 0.4017 40.5459
s.e. 0.0499 0.0522 0.0009 0.2142
sigma^2 estimated as 8.053: log likelihood=-982.81
AIC=1975.62 AICc=1975.77 BIC=1995.58
Series: value
Model: ARIMA(3,1,0) w/ drift
Coefficients:
ar1 ar2 ar3 constant
-0.7765 -0.5859 -0.2664 1.0492
s.e. 0.0482 0.0545 0.0482 0.1640
sigma^2 estimated as 10.78: log likelihood=-1038.92
AIC=2087.84 AICc=2087.99 BIC=2107.78
```

tS is a deterministic trend stationary process, thus the first model should be appropriate and the second model not, correct? How to interpret i.e. what do the results and coefficients mean, from both models?

If I change the seed e.g. set.seed(97523) the tS series (from a distance) looks very similar - if I re-estimate, I get very different results. Why?

In fable:

[1] What is the correct way to model and generate multistep OOS forecasts including prediction intervals for the tS process?

(a) Why can I not just model the tS process with automatic ARIMA or ETS?

(b) Another way could be to detrend, i.e. remove the trend with lm, and then estimate and generate multistep OOS forecasts for the residuals with ARIMA or ETS, and then add the forecast trend back. How in fable including prediction intervals?

[2] If I add a pure RW term to tS, what is the appropriate way to model the tSxt process?

```
xt <- arima.sim(n = 500, n.start = 100, list(order=c(0,1,0)))
xt
tSxt <- tail(tS,400) + tail(xt,400)
tSxt
plot.ts(tSxt[1:(nburn+nobs)])
```

thanks,

Amarjit