The Hodrick-Prescott filter is a mathematical tool used in macroeconomics, specifically in the analysis of business cycles. It is used to obtain a smoothed non-linear representation of a time series, one that is more sensitive to long-term than to short-term fluctuations. The adjustment of the sensitivity of the trend to short-term fluctuations is achieved by modifying a multiplier λ.
The formula
The reasoning for the formula is as follows: Let 
for 
denote the logarithms of a time series variable. Given an adequately chosen, positive value of λ, there is a "trend component", denoted by 
, that minimizes
![\sum_{t = 1}^T {(y_t - \tau _t )^2 } + \lambda \sum_{t = 2}^{T - 1} {[(\tau _{t+1} - \tau _t) - (\tau _t - \tau _{t - 1} )]^2 }.\,](a/../upload/math/302de546828ebee8262fe6ea95501bcc.png)
The first term of the equation is the sum of the squared deviations dt = yt - τt. The second term is a multiple λ of the sum of the squares of the trend component's second differences. This second term penalizes variations in the growth rate of the trend component. The larger the value of λ, the higher is the penalty. Hodrick and Prescott advise that, for quarterly data, a value of λ = 1600 is reasonable.
The filter was first applied by economists Robert J. Hodrick and recent Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel winner Edward C. Prescott. Though Hodrick and Prescott were the first to make use of the filter in the field of economics, it is believed that others, such as the mathematician John von Neumann, had already employed similar versions in the past.
