Sign Accuracy, Mean-Squared Error and the Rate of Zero Crossings: a Generalized Forecast Approach
Marc Wildi
TL;DR
The paper tackles the forecast-design dilemma of balancing accuracy and smoothness by introducing Smooth Sign Accuracy (SSA), a framework that jointly considers sign accuracy, mean-squared error (MSE), and the rate of sign changes in the predictor. SSA formalizes a constrained optimization that maximizes a target correlation with the future target while enforcing a holding-time constraint and a fixed predictor length, yielding a one-parameter family of optimal predictors $\mathbf{b}(\nu)$ that interpolate between maximal smoothing and maximal responsiveness. It provides a frequency-domain interpretation via SSA-AR(2) filters and a time-domain difference-equation view, establishing a bijection between HT and the first-order autocorrelation under Gaussianity. The work extends SSA to non-stationary (I(1), I(2)) settings with cointegration constraints, develops extensions for incomplete spectral support, and demonstrates practical utility through benchmark customization (HP, Hamilton, Baxter-King) and a real-world application to US-INDPRO trend-nowcasting. Overall, SSA offers a versatile, interpretable framework that improves forecast performance while controlling the occurrence of sign changes, enabling tailored, robust real-time forecasting in economics and beyond.
Abstract
Forecasting entails a complex estimation challenge, as it requires balancing multiple, often conflicting, priorities and objectives. Traditional forecast optimization criteria typically focus on a single metric -- such as minimizing the mean squared error (MSE) -- which may overlook other important aspects of predictive performance. In response, we introduce a novel approach called the Smooth Sign Accuracy (SSA) framework, which simultaneously considers sign accuracy, MSE, and the frequency of sign changes in the predictor. This addresses a fundamental trade-off (the so-called accuracy-smoothness (AS) dilemma) in prediction. The SSA criterion thus enables the integration of various design objectives related to AS forecasting performance, effectively generalizing conventional MSE-based metrics. We further extend this methodology to accommodate non-stationary, integrated processes, with particular emphasis on controlling the predictor's monotonicity. Moreover, we demonstrate the broad applicability of our approach through an application to, and customization of, established business cycle analysis tools, highlighting its versatility across diverse forecasting contexts.