Time Series Filters: Options

Time Series Filters dialog contains the following options.

Hodrick-Prescott (HP)

Hodrick-Prescott (HP) filter is a tool used in macroeconomics for fitting a smooth trend to time series.

The Hodrick-Prescott filter separates a time-series y_t into a trend component τ_t and a cyclical component c_t.

y_t=τ_t+c_t+ϵ_t where, ϵ_tis the error (or noise) component.

The components are determined by minimizing the quadratic loss function and the HP filter is explicitly given by the formula,

HP = [λL²− 4λL+ (1 + 6λ)− 4λL⁻¹ + λL⁻²]⁻¹

where λ denotes the smoothing parameter (positive value) and L denotes the lag operator, which is derived from the first-order condition for the minimization problem.

Choose one variable for HP filter from the list of variables and set a value for Lambda.

The smoothing parameter λ determines the balance between fit and smoothness. The larger the value of λ, the higher is the penalty. When you analyze quarterly data, the default value of 1600 is recommended for λ. Also, λ should be 1600/4⁴ = 6.25 for annual data, and 1600 ∗ 3⁴ = 129,600 for monthly data. When λ= 0, it means that the output of the HP filter series matches the input series exactly. On the contrary, higher values of λ yield smoother output series.

Baxter-King (BK)

Baxter-King (BK) is a band-pass filter commonly used in time series analysis to isolate cyclical components of data, often in macroeconomic studies. Its goal is to remove high-frequency (short-term) noise and low-frequency (long-term trend) components, focusing on the medium-term cycles of interest, typically between 1.5 and 8 years. The BK filter assumes stationarity of the underlying time series. From the list of variables, choose 1 or 2 variables for BK filter.

Set appropriate values in Low (minimum period), High (maximum period) and K (Lead-lag length).

Christiano-Fitzgerald (CF)

The Christiano-Fitzgerald (CF) filter is another band-pass filter used to separate a time series into trend and cyclical components. This filter is similar to the Baxter King (BK) filter, which also works on the basis of an ideal band-pass filter where the BK filter assumes a symmetric moving average to approximate the band-pass filter, but the CF filter assumes that the time series is following randomness without the drift. CF filtering is implemented by estimating an optimal filter under the assumption that the data is generated by a random walk or ARIMA-type process.

From the list of variables, choose 1 or 2 variables for CF filter. Set appropriate values in Low, High, and K. Select Drift if needed for CF filter.