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Holt's linear trend and the damped trend

Today let's understand Holt's linear trend method, a way to forecast a series that keeps climbing year after year, not one that just sits flat.

Take the number of international airline passengers, counted every month from 1949 to 1960 and added up into 12 yearly totals, in thousands of passengers a year. Here is all 12 years of it, plotted in order.

Look at that line. It climbs every single year, from 1520 thousand in 1949 to 5714 thousand in 1960, without a single year of falling back. That is exactly the shape Holt's linear trend method was built to forecast.

Why a flat forecast fails a series that keeps climbing

Every one of those 12 years rose over the year before it. So the natural next question is: what does the simplest forecasting method, one with no trend at all, do with a series shaped like this?

Build the 12 annual totals as a tsibble, fit a model with no trend, and forecast three years ahead.

RInteractive R
# Build 12 annual totals from the monthly AirPassengers series, then fit a model with no trend library(fable) library(tsibble) library(dplyr) annual <- tapply(as.numeric(AirPassengers), rep(1949:1960, each = 12), sum) air <- tsibble(Year = 1949:1960, Passengers = as.numeric(annual), index = Year) fit_ses <- air |> model(ETS(Passengers ~ error("A") + trend("N") + season("N"))) forecast(fit_ses, h = 3) |> as.data.frame() |> transmute(Year, forecast = round(.mean, 0)) #> Year forecast #> 1 1961 5714 #> 2 1962 5714 #> 3 1963 5714

  

  





The forecast for 1961, 1962 and 1963 is the same number three times over, 5714, the last year's total, repeated forever. Passengers had grown every single year for over a decade, and this model just throws that growth away.



That is simple exponential smoothing at work: it carries a level forward and nothing else, so once the last real number arrives, the forecast just repeats it. Holt's linear trend method, first described by Charles Holt in 1957, fixes exactly this. Alongside the level, it tracks a second smoothed number, the trend, that captures how much the series has been rising or falling, and carries that rise forward into the forecast too.