Fan chart (time series)

Predicted future values can be diagrammed in various ways; most simply, by a single predicted value, and an upper and lower range around that (three lines total), or by various future intervals, depicted by varying degrees of shading (darkest near the center of the range, fainter near the ends of the range).

There are several ways to represent the forecast density depending on the shape of the forecasting distribution.

• If the forecast density is symmetric (normal or Student's t, for instance), the fan centers at the mean (which coincides with the mode and median) forecast, and the ranges expand like confidence intervals by adding and subtracting multiples of the forecasting standard error to the mean forecast. These ranges are known as equal-tail ranges and centre at the mean forecast. Low resolution charts may add and subtract one, two and three forecasting standard errors for approximate coverages of 68%, 95% and 99.7%. These charts can easily be built through standard Excel graphs.
• If the forecast density is non-symmetric, centering the fan at the median and using equal tail ranges might not be appropriate as it would overstate the forecast uncertainty. In this case it is better to center the fan at the more likely forecast (the mode) and use Highest Probability Density (HPD) ranges.^[3] HPDs are by definition the shortest ranges covering a given probability, say 50%, and are centered at the mode. In this case it is usual to include increasing probability ranges of 10%, 20%, ..., 90%, for instance.^[4][5]

In the Bank of England's implementation it is assumed that the forecast distribution is a two-piece normal or split normal density.^[6] This density results from joining the two-halves of corresponding normal densities with the same mode but different variances. As a result, the split normal density is non-symmetric and uni-modal. In this case, inflation forecast fan charts are usually accompanied with the balance of risks, the probability that the future inflation falls below its modal forecast. In this way, central banks that employ inflation targeting report to the general public not only the more likely forecasts of the inflation rate but also its balance of risks!^[7]

The split normal density is completely characterized by three parameters, the mode, variance and skewness. Therefore, the fan chart ranges depend on these parameters only.^[4][5][6] and^[8]

In a central bank that employs inflation targeting, the three moments of the inflation forecast distribution are determined as follows:

• The mode. Modal forecasts are derived from the suite of models of the central bank.
• The variance. Standard errors of forecasts might be derived from appropriately formulated forecasting models but it is advisable to derive them from historical forecasting errors instead.^[2]
• The skewness. A mapping from the skewness (or balances of risks) of factors that affect the inflation rate along the forecast horizon to the skewness of the inflation forecast distribution has to be specified.^[7]

• Inflation Report Fan Charts Archived 7 July 2016 at the Wayback Machine, Bank of England
• Fan chart, MATLAB Central, by Marco B., 23 May 2010
• fanplot, CRAN R Package by Guy J. Abel, 9 April 2013