Hodrick-Prescott (HP) Filter

The Hodrick-Prescott (HP) filter is a statistical procedure that is used to decompose a time series into a trend and a cyclical component. It is named after its inventors, Robert Hodrick and Edward Prescott.

The HP filter is a nonlinear filter that is based on the assumption that the trend component of a time series is smooth, while the cyclical component is more irregular. The filter works by minimizing the sum of squared deviations between the original time series and a smoothed version of the time series. The smoothing is done using a weighted moving average, where the weights are determined by the HP filter's smoothing parameter.

The HP filter is often used to decompose economic time series, such as gross domestic product (GDP), industrial production, and employment. The trend component of the time series can be used to assess long-term economic growth, while the cyclical component can be used to identify economic cycles.

The HP filter has several advantages over other methods of decomposing time series. First, it is relatively simple to implement. Second, it is robust to outliers and noise. Third, it does not require any assumptions about the underlying distribution of the time series.

However, the HP filter also has some limitations. First, it is not always clear how to choose the smoothing parameter. Second, the filter can sometimes produce spurious results, such as negative trend components or cyclical components that are too large.

Despite these limitations, the HP filter is a widely used and effective method of decomposing time series. It is a valuable tool for economists and other analysts who are interested in understanding the long-term and cyclical trends in economic activity.

Here is a more mathematical explanation of the HP filter. The HP filter is a linear filter that is defined by the following equation:

$$y_t = \alpha y_{t-1} + (1-\alpha)x_t$$

where $y_t$ is the filtered time series, $x_t$ is the original time series, and $\alpha$ is the smoothing parameter. The smoothing parameter controls the amount of smoothing that is applied to the time series. A small value of $\alpha$ will result in a more smooth trend component, while a large value of $\alpha$ will result in a more irregular trend component.

The HP filter can be used to decompose a time series into a trend and a cyclical component. The trend component is given by the following equation:

$$\hat{g}_t = \sum_{i=0}^{\infty} \alpha^i x_{t-i}$$

The cyclical component is given by the following equation:

$$\hat{c}_t = x_t - \hat{g}_t$$

The HP filter is a powerful tool that can be used to analyze economic time series. It is a valuable tool for economists and other analysts who are interested in understanding the long-term and cyclical trends in economic activity.