`hpf` (Series)

Hodrick-Prescott filter with conditioning information

Syntax

[T, C, cutoff, lambda] = hpf(X, ...)

Syntax with output arguments swapped

[C, T, cutoff, lambda] = hpf2(X, ...)

Input arguments

x [ Series ]

Input time series that will be filtered.

Output arguments

t [ Series ]

Low-frequency (trend) component.

c [ Series ]

High-frequency (cyclical or gap) component.

cutoff [ numeric ]

Cut-off periodicity; periodicities above the cut-off are attributed to trends, periodicities below the cut-off are attributed to gaps.

lambda [ numeric ]

Smoothing parameter actually used; this output argument is useful when lambda=@auto or when the option cutoff= is used instead of lambda=.

Options

cutoff=[ ] [ numeric ]

Cut-off periodicity in periods (depending on the time series frequency); this option can be specified instead of lambda=; the smoothing parameter will be then determined based on the cut-off periodicity.

cutoffYear=[ ] [ numeric ]

Cut-off periodicity in years; this option can be specified instead of lambda=; the smoothing parameter will be then determined based on the cut-off periodicity.

gamma=1 [ numeric | Series ]

Weight or weights on the deviations of the trend from observations; it only makes sense to use this option to make the signal-to-noise ratio time-varying; see the optimization problem below.

infoSet=2 [ 1 | 2 ]

Information set assumption used in the filter: 1 runs a one-sided filter, 2 runs a two-sided filter.

lambda=@auto [ numeric | @auto ]

Smoothing parameter; needs to be specified for Series objects with indeterminate frequency; see Description for default values.

level= [ Series ]

Time series with hard tunes and soft tunes on the level of the trend.

change= [ Series ]

Time series with hard tunes and soft tunes on the change in the trend.

log= [ true | false ]

Logarithmize the data before filtering, de-logarithmize afterwards.

Description

The underlying optimization problem

The function hpf solves a constrained optimization problem described by the following Lagrangian

\[ \min_{\bar y_t, \omega_t, \sigma_t} \underbrace{ \sum \lambda \left( \Delta \bar y_t - \Delta \bar y_{t-1} \right)^2 + \sum \gamma_t \left( \bar y_t - y_t \right)^2}_\text{Plain HP with time-varying signal-to-noise ratio} + \cdots \]

\[ \cdots + \underbrace{\sum u_t \left( \bar y_t - a_t \right)^2}_\text{Soft level tunes} + \underbrace{\sum v_t \left( \Delta \bar y_t - b_t \right)^2}_\text{Soft growth tunes} + \underbrace{\sum \omega_t \left( \bar y_t - c_t \right)}_\text{Hard level tunes} + \underbrace{\sum \sigma_t \left( \Delta \bar y_t - d_t \right)}_\text{Hard growth tunes} \]

where

\(\(\Delta\)\) is the first-difference operator;
\(\(\lambda\)\) is a (scalar) smoothing parameter;
\(\(y_t\)\) are user-supplied observations;
\(\(\bar y_t\)\) is the fitted trend;
\(\(\gamma_t\)\) are user-supplied weights to modify the basic signal-to-noise ratio over time (the default setting is \(\(\gamma_t=1\)\)), entered in the option gamma=;
\(\(a_t\)\) and \(\(u_t\)\) are soft tunes on the level of the trend and the weights associated with these soft level tunes, respectively, entered together as complex numbers in the option level=;
\(\(b_t\)\) and \(\(v_t\)\) are soft tunes on the change in the level of the trend and the weights associated with these soft growth tunes, respectively, entered together as complex numbers in the option change=;
\(\(c_t\)\) are hard tunes on the level of the trend, entered as real numbers in the option level=;
\(\(d_t\)\) are hard tunes on the change in the level of the trend, entered as real numbers in the option change=;
\(\(\omega_t\)\) are Lagrange multipliers on the hard level tunes (note that these are computed as part of the optimization problem, not entered by the user);
\(\(\sigma_t\)\) are Lagrange multipliers on the hard growth tunes (note that these are computed as part of the optimization problem, not entered by the user).

Each of the summations in the above Lagrangian goes over those periods in which the respective bracketed terms are defined (observations or tunes exist). You can combine any number of any tunes in one run of hpf, including out-of-sample tunes (see below).

Imposing tunes on trend level and trend change

The hard tunes and soft tunes on the level of the trend are entered as time series through the option level=.
The hard tunes and soft tunes on the change in the trend are entered as time series through the option change=.
In the time series entered through level= and/or change=, you can combine any number of hard and soft tune. In each particular period, you can obviously specify only a hard tune or only a soft tune. You can think of hard tunes as a special case of soft tunes with infinitely large weights.
A hard tune is specified as a plain real number (i.e. a number with a zero complex part).
A soft tune must be entered as a complex number whose real part specifies the tune itself, and the imaginary part specifies the inverse of the weight, i.e. \(\(1/v_t\)\) or \(\(1/u_t\)\), on that tune in that period. Note that if the weight goes to infinity, the imaginary part becomes zero and the tune becomes a hard tune.

Out-of-sample tunes

Tunes can be imposed also at dates before the first observation of the input series, or after the last observation. In other words, the time series in level= and/or change= can have a more extended range (at either side) than the filtered input series.

Default smoothing parameters

If the smoothing parameter lambda= is not specified (i.e. lambda=@auto), a default value is computed. The default value is based on common practice and can be calculated using the date frequency of the input time series as \(\(\lambda = 100 f^2\)\), where \(\(f\)\) is the frequency (yearly=1, half-yearly=2, quarterly=4, monthly=12). This gives the following default values:

100 for yearly time series (cut-off periodicity of 19.79 years);
400 for half-yearly time series (cut-off periodicity of 14.02 years);
1,600 for quarterly time series (cut-off periodicity of 9.92 years);
3,600 for bi-monthly time series (cut-off periodicity of 8.11 years);
14,400 for monthly time series (cut-off periodicity of 5.73 years).

Note that there is no default value for data with indeterminate or daily frequency: for these types of time series, you must always use the option lambda=.

hpf (Series)