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Hodrick-Prescott filter for trend and cyclical components

`hpfilter(S)hpfilter(S,smoothing)T = hpfilter(...)[T,C] = hpfilter(...)`

`hpfilter(S)`uses a Hodrick-Prescott filter and a default smoothing parameter of 1600 to separate the columns of`S`into trend and cyclical components.`S`is an*m*-by-*n*matrix with*m*samples from*n*time series. A plot displays each time series together with its trend (the time series with the cyclic component removed).`hpfilter(S,smoothing)`applies the smoothing parameter`smoothing`to the columns of`S`. If`smoothing`is a scalar,`hpfilter`applies it to all columns. If`S`has*n*columns and`smoothing`is a conformable vector (*n*-by-1 or 1-by-*n*),`hpfilter`applies the vector components of`smoothing`to the corresponding columns of`S`.If the smoothing parameter is

`0`, no smoothing takes place. As the smoothing parameter increases in value, the smoothed series becomes more linear. A smoothing parameter of`Inf`produces a linear trend component.Appropriate values of the smoothing parameter depend upon the periodicity of the data. The following reference suggests the following values:

Yearly — 100

Quarterly — 1600

Monthly — 14400

`T = hpfilter(...)`returns the trend components of the columns of`S`in`T`, without plotting.`[T,C] = hpfilter(...)`returns the cyclical components of the columns of`S`in`C`, without plotting.

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