Filters Refresher — ECON-5371

Simulated Data-Generating Process

y(t) = trend(t) + cycle(t) + noise(t). Because this is simulated, the true trend is known — the one advantage a textbook example on real GDP can't offer.

Components

Trend curvature 0.60

Cycle amplitude 2.0

Cycle period (quarters) 32

Noise volatility 1.0

Show Filters

True trend (ground truth)

HP filter (λ=1600)

Linear detrend

Kalman / local level (smoothed)

ETS / Holt level

HP End-Point Bias Demo

Full-sample HP trend Real-time HP trend (re-run each period)

Fit Quality vs. True Trend

Extracted Trends vs. Ground Truth

Observed y(t) True trend HP filter Linear detrend Kalman (smoothed) ETS / Holt

Extraction Error by Method (RMSE vs. true trend)

HP End-Point Bias: Full-Sample vs. Real-Time Trend (last 40 periods)