ETS trend from components() is different to that calculated using the trend formula - if alpha and beta parameters are both set to 0.5

[PhilBla]

Not absolutely sure this is a bug but can't figure out why the Holt linear trend equations produces different slope values to those given by the components() view of an ETS model. I have posted a reproducible example on Cross Validated

https://stats.stackexchange.com/questions/630437/i-cannot-replicate-the-slope-for-an-ets-model-with-alpha-0-5-and-beta-0-5-usi

I am not skilled enough to look through the code, but hope that you can confirm that it should be possible to replicate the level and trend using the Holt equations.

Philip

[robjhyndman]

The equation in your post uses beta*. The function uses beta. See https://otexts.com/fpp3/ets.html#etsaan-holts-linear-method-with-additive-errors for the connection

[PhilBla]

Thanks very much Rob for taking the time to respond to my question. It is a steep learning curve, and I hope you have the patience to answer one more. Having spent time rewatching and rereading chapter 8.5 and some more time reevaluating the formulas, I remain unable to calculate the first estimated slope value using the components function values and β=αβ*

Q. So if the ETS model is specified with parameters: trend(method = "A", alpha = 0.5, beta = 0.5))
is α = alpha and β* = beta ? or is β = beta ?

... to calculate the new estimated slope the equation is: bt = bt−1 + βεt - where β=αβ*
which (if β* = beta) becomes bt-1 + (0.5*0.5) * εt

If I use 0.5*0.5 I can't get the same answer as the components function. If I just use 0.5 it works. So my conclusion is β (without the *) must be equivalent to the trend parameter beta.

Code
library(fpp3)
aus_economy <- global_economy |> filter(Code == "AUS") |> mutate(Pop = Population / 1e6)

fit <- aus_economy |>
model(AAN = ETS(Pop ~ error("A") + trend(method = "A", alpha = 0.5, beta = 0.5) + season("N")))
report(fit)

options(pillar.sigfig = 10)
summary <- components(fit) |> left_join(fitted(fit), by = c("Country", ".model", "Year"))
summary

CalculatedSlope <- summary$slope[1] + (0.5 * summary$remainder[2])
CalculatedSlope - summary$slope[2] # CalculatedSlope is correct - so β must be = beta

CalculatedSlope <- summary$slope[1] + ((0.5 *0.5) * summary$remainder[2])
CalculatedSlope - summary$slope[2] # CalculatedSlope is different

[robjhyndman]

The package always uses beta (not beta*).

The first estimate of slope (at time 0) is optimized. You can't compute it from the other components.