2006 AnotherLookAtMeasuresOfForecastAccuracy

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Subject Headings: Forecast accuracy; Forecast evaluation; Forecast error measures; M-competition; Mean absolute scaled error

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Author Keywords

Forecast accuracy; Forecast evaluation; Forecast error measures; M-competition; Mean absolute scaled error

Abstract

We discuss and compare measures of accuracy of univariate time series forecasts. The methods used in the M-competition as well as the M3-competition, and many of the measures recommended by previous authors on this topic, are found to be degenerate in commonly occurring situations. Instead, we propose that the mean absolute scaled error become the standard measure for comparing forecast accuracy across multiple time series.

2 A critical survey of accuracy measures

Let [math]\displaystyle{ Y_t }[/math] denote the observation at time [math]\displaystyle{ t }[/math] and [math]\displaystyle{ F_t }[/math] denote the forecast of [math]\displaystyle{ Y_t }[/math]. Then define the forecast error [math]\displaystyle{ e_t = Y_t - F_t }[/math]. The forecasts may be computed from a common base time, and be of varying forecast horizons. Thus, we may compute out-of-sample forecasts [math]\displaystyle{ Fn+1, . . . , Fn+m }[/math] based on data from times [math]\displaystyle{ t = 1,...,n }[/math]. Alternatively, the forecasts may be from varying base times, and be of a consistent forecast horizon. That is, we may compute forecasts [math]\displaystyle{ F_{1+h},...,F_{m+h} }[/math] where each [math]\displaystyle{ F_{j+h} }[/math] is based on data from times [math]\displaystyle{ t=1,...,j }[/math]. The in-sample forecasts in the examples above were based on the second scenario with [math]\displaystyle{ h=1 }[/math]. A third scenario arises when we wish to compare the accuracy of methods across many series at a single forecast horizon. Then we compute a single [math]\displaystyle{ F_{n+h} }[/math] based on data from times [math]\displaystyle{ t=1,...,n }[/math] for each of m different series.

We do not distinguish these scenarios in this paper. Rather, we simply look at ways of summarizing forecast accuracy assuming that we have m forecasts and that we observe the data at each forecast period.

We use the notation mean([math]\displaystyle{ x_t }[/math]) to denote the sample mean of [math]\displaystyle{ {x_t} }[/math] over the period of interest (or over the series of interest). Analogously, we use median([math]\displaystyle{ x_t }[/math]) for the sample median and gmean([math]\displaystyle{ x_t }[/math]) for the geometric mean.

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