Fan chart (time series)
Encyclopedia
In time series
analysis, a fan chart is a chart that joins a simple line chart
for observed past data, by showing ranges for possible values of future data together with a line showing a central estimate or most likely value for the future outcomes. As predictions become increasingly uncertain the further into the future one goes, these forecast ranges spread out, creating distinctive wedge or "fan" shapes, hence the term. Alternative forms of the chart can also include uncertainty for past data, such as preliminary data that is subject to revision.
The term "fan chart" was coined by the Bank of England
, which has been using these charts and this term since 1997 in its "Inflation Report" to describe its best prevision of future inflation
to the general public. Fan charts have been used extensively in finance
and monetary policy
, for instance to represent forecasts of inflation
.
There are several ways to represent the forecast density depending on the shape of the forecasting distribution.
In the Bank of England’s implementation it is assumed that the forecast distribution is a two piece normal or split normal density
. 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!
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. and
In a central bank that employs inflation targeting
, the three moments of the inflation forecast distribution are determined as follows:
Time series
In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
analysis, a fan chart is a chart that joins a simple line chart
Line chart
A line chart or line graph is a type of graph, which displays information as a series of data points connected by straight line segments. It is a basic type of chart common in many fields. It is an extension of a scatter graph, and is created by connecting a series of points that represent...
for observed past data, by showing ranges for possible values of future data together with a line showing a central estimate or most likely value for the future outcomes. As predictions become increasingly uncertain the further into the future one goes, these forecast ranges spread out, creating distinctive wedge or "fan" shapes, hence the term. Alternative forms of the chart can also include uncertainty for past data, such as preliminary data that is subject to revision.
The term "fan chart" was coined by the Bank of England
Bank of England
The Bank of England is the central bank of the United Kingdom and the model on which most modern central banks have been based. Established in 1694, it is the second oldest central bank in the world...
, which has been using these charts and this term since 1997 in its "Inflation Report" to describe its best prevision of future inflation
Inflation
In economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time.When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation also reflects an erosion in the purchasing power of money – a...
to the general public. Fan charts have been used extensively in finance
Finance
"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...
and monetary policy
Monetary policy
Monetary policy is the process by which the monetary authority of a country controls the supply of money, often targeting a rate of interest for the purpose of promoting economic growth and stability. The official goals usually include relatively stable prices and low unemployment...
, for instance to represent forecasts of inflation
Inflation
In economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time.When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation also reflects an erosion in the purchasing power of money – a...
.
Implementation
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 modeMode (statistics)In statistics, the mode is the value that occurs most frequently in a data set or a probability distribution. In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score....
and medianMedianIn probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...
) forecast, and the ranges expand like confidence intervalConfidence intervalIn statistics, a confidence interval is a particular kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval , in principle different from sample to sample, that frequently includes the parameter of interest, if the...
s 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. 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.
In the Bank of England’s implementation it is assumed that the forecast distribution is a two piece normal or split normal density
Split normal distribution
In probability theory and statistics, the split normal distribution also known as the two-piece normal distribution results from joining at the mode the corresponding halves of two normal distributions with the same mode but different variances...
. 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
Inflation targeting
Inflation targeting is an economic policy in which a central bank estimates and makes public a projected, or "target", inflation rate and then attempts to steer actual inflation towards the target through the use of interest rate changes and other monetary tools.Because interest rates and the...
report to the general public not only the more likely forecasts of the inflation rate but also its balance of risks!
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. and
In a central bank that employs inflation targeting
Inflation targeting
Inflation targeting is an economic policy in which a central bank estimates and makes public a projected, or "target", inflation rate and then attempts to steer actual inflation towards the target through the use of interest rate changes and other monetary tools.Because interest rates and the...
, 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.
- 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.
External links
- Inflation Report Fan Charts, Bank of England
- Fan chart, MATLAB Central, by Marco B., 23 May 2010