t=1

Macroeconomic time series exhibit trends, are therefore not stationary and therefore have no well-defined moments. Growth rates: extremely short cycles. Deviations from linear trend: extremely long cycles. More flexible: the HP filter. Hodrick and Prescott (1980). Apparently it originated in the engineering literature, where it is known as the Whitaker–Henderson filter. The idea is to generalize the linear trend and allow the slope of the “trend ” to vary — but not too much. Formally, we trade off smoothness of the trend (closeness to linearity) and close tracking of the series. By definition, the HP trend X̂t solves min ⟨X̂t⟩T t=0 T∑ t=