# Time horizon in forecasting, and rules of thumb

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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 = AirPassengers ETS = 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,

or see also here

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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