Trend Estimation and the Hodrick-prescott Filter

The irregular and slope disturbances, εt and ζt, respectively, are mutually independent and the notation NID(0, σ2) denotes normally and independently distributed with mean zero and variance σ2. The signal-noise ratio, q = σ2 ζ/σ 2 ε , plays the key role in determining how observations should be weighted for prediction and signal extraction. The higher is q, the more past observations are discounted in forecasting the future. Similarly a higher q means that the closest observations receive a bigger weight when signal extraction is carried out. The trend is known as an integrated random walk trend. When estimated it tends to be relatively smooth and indeed fitting the continuous time version of the model is known to be equivalent to fitting a cubic spline; see Wecker and Ansley (1983). The statistical treatment of such unobserved component models is based on the state space form (SSF), as described in Harvey (1989) and Kitagawa and Gersch (1996). Once a model has been put in SSF, the Kalman filter yields