# Time horizon in forecasting, and rules of thumb

April 7, 2011
By

(This article was first published on Freakonometrics - Tag - R-english, and kindly contributed to R-bloggers)

I recently received an email about forecasting and rules of thumb. “Dans
la profession
[…] se
transmet une règle empirique qui voudrait que l’on prenne un historique
du double de l’horizon de prévision : 20 ans de données pour une
prévision à 10 ans, etc… Je souhaite savoir si cette règle n’aurait
pas, par hasard, un fondement théorique quitte à ce que le rapport ne
soit pas de 2 pour 1, mais de 3 pour 1 ou de 1 pour 1 par exemple.

To summarize briefly, the rule is to consider a 2-1 ratio for the
period of observation vs. forecast horizon. And the interesting
question is if there are justifications for such a rule…

At first, I remembered a rules of thumb, from the book by Box and Jenkins, which states that it is meaningless to look at autocorrelations when lags exceed the sample size over 6. So with 12 years of data, autocorrelations with a lag higher than two years are useless. But it is not what is mentioned here. So I looked at some dataset, and some standard time series models.

• It
depends on the series

It might obvious… but if it is the case, it means that it will be
difficult to have a general
rule of thumb. Consider e.g. the number of airline passengers,

`library(forecast)X = AirPassengersETS = ets(X)plot(forecast(ETS,h=length(X)/2))`

or some sales in a big store,

or car casualties in France, or the temperature in Nottingham Castle,

or the water level at Lack Hurron, or the flow of the Nile river,

for forecasting techniques in demography. Actually, in the case of life
insurance, actuaries have to forecast future demography, i.e. try to
assess death rates of those who currently purchase retirement
contracts, who might be 20 years old. So they have to forecast death
rate until 2100, say. One the one hand, it sounds difficult to make
forecast over a century (it is already difficult for climate, I guess
it is even more complex for human life). On the other hand, a 2-1 ratio
means that we have to use data from 1800… Here again, it is difficult
to justify that mortality in the 1850 could be interesting to say
anything about mortality in 2050. So I guess it will be difficult to
justify the use of general rules of thumb….

• It
depends on the model

Consider the following (simulated) series. Several models can be
fitted. And the shape on the forecast (and the forecast error) will
depend on the model considered. The benchmark
can be the model without any dynamics, i.e. we assume that observations
are i.i.d. Or more classically, assume that it is simple a white noise, i.e. an i.i.d centered
process. Then the forecast is the following,

With that kind of assumption, we see that the 2-1 ratio is useless since
we can get forecasts up to any horizon…. But that does not seem
very robust. For instance, if we consider exponential smoothing techniques, we can
obtain

Which is rather different. And with the 2-1 ratio, obviously, there is
a lot of uncertainty at the end ! It would be even worst if we assume
that we look at a random walk. Because actually a dozen models – at
least – can be considered, from ARIMA, seasonal ARIMA, Holt Winters,
Exponential Smoothing, etc…

So I do not see any theoretical justification of that rule of thumb. Obviously, the maximum horizon can not be extremely far away if the
series is non-stationary, with a very irregular pattern, and with a lot
of noise… So we’re back at the beginning. If anyone is willing to
share his or her experience, comments are open.

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