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The Accuracy Smoothness Dilemma in Prediction: a Novel Multivariate M-SSA Forecast Approach

Marc Wildi

TL;DR

The recently developed Smooth Sign Accuracy (SSA) framework extends the traditional MSE approach by simultaneously accounting for sign accuracy, MSE, and the frequency of sign changes in the predictor, effectively generalizing conventional MSE-based metrics.

Abstract

Forecasting presents a complex estimation challenge, as it involves balancing multiple, often conflicting, priorities and objectives. Conventional forecast optimization methods typically emphasize a single metric--such as minimizing the mean squared error (MSE)--which may neglect other crucial aspects of predictive performance. To address this limitation, the recently developed Smooth Sign Accuracy (SSA) framework extends the traditional MSE approach by simultaneously accounting for 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. We extend this approach to the multivariate M-SSA, leveraging the original criterion to incorporate cross-sectional information across multiple time series. As a result, the M-SSA criterion enables the integration of various design objectives related to AS forecasting performance, effectively generalizing conventional MSE-based metrics. To demonstrate its practical applicability and versatility, we explore the application of the M-SSA in three primary domains: forecasting, real-time signal extraction (nowcasting), and smoothing. These case studies illustrate the framework's capacity to adapt to different contexts while effectively managing inherent trade-offs in predictive modelling.

The Accuracy Smoothness Dilemma in Prediction: a Novel Multivariate M-SSA Forecast Approach

TL;DR

The recently developed Smooth Sign Accuracy (SSA) framework extends the traditional MSE approach by simultaneously accounting for sign accuracy, MSE, and the frequency of sign changes in the predictor, effectively generalizing conventional MSE-based metrics.

Abstract

Forecasting presents a complex estimation challenge, as it involves balancing multiple, often conflicting, priorities and objectives. Conventional forecast optimization methods typically emphasize a single metric--such as minimizing the mean squared error (MSE)--which may neglect other crucial aspects of predictive performance. To address this limitation, the recently developed Smooth Sign Accuracy (SSA) framework extends the traditional MSE approach by simultaneously accounting for 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. We extend this approach to the multivariate M-SSA, leveraging the original criterion to incorporate cross-sectional information across multiple time series. As a result, the M-SSA criterion enables the integration of various design objectives related to AS forecasting performance, effectively generalizing conventional MSE-based metrics. To demonstrate its practical applicability and versatility, we explore the application of the M-SSA in three primary domains: forecasting, real-time signal extraction (nowcasting), and smoothing. These case studies illustrate the framework's capacity to adapt to different contexts while effectively managing inherent trade-offs in predictive modelling.
Paper Structure (16 sections, 7 theorems, 79 equations, 14 figures, 6 tables)

This paper contains 16 sections, 7 theorems, 79 equations, 14 figures, 6 tables.

Table of Contents

  1. Introduction
  2. M-SSA Criterion
  3. Univariate SSA
  4. SSA Interpretations
  5. Multivariate M-SSA
  6. Dependence
  7. Embedding, Smoothing and Efficient Frontier
  8. Solution
  9. Applications
  10. Forecasting
  11. Smoothing
  12. Signal Extraction
  13. Univariate: HP-C and SSA
  14. M-SSA: Bivariate Leading Indicator Design
  15. Conclusion
  16. ...and 1 more sections

Key Result

Proposition 1

Under the assumptions of white noise and a full-rank covariance matrix $\boldsymbol{\Sigma}$ as established in the previous section, the vector $\mathbf{b}_i$ of the M-SSA Criterion mcrit1 constitutes a stationary point of the first-order autocorrelation $\rho(y_{i})$ if and only if $\mathbf{b}_{ij}

Figures (14)

  • Figure 1: M-SSA (top) and MSE (bottom) predictors for the first target series $z_{1t+\delta}$ (left) and the second target series $z_{2t+\delta}$ (right) with $\delta=1$ (one-step ahead). Predictor weights applied to first series $z_{1t}$ (blue) and second series $z_{2t}$ (red). The weights of the MSE predictor correspond to the first row of $A_1$ (bottom left) and the second row of $A_1$ (bottom right), respectively.
  • Figure 2: A comparison of filter outputs: M-SSA (cyan), MSE (violet) and target (black) for $\delta=1$. Zero-crossings by M-SSA are marked by vertical lines.
  • Figure 3: SSA smoothers scaled to unit length as a function of $\delta$ with values ranging from $\delta=-(L-1)/2=-100$ (symmetric filter) to $\delta=0$ (asymmetric nowcast), assuming the data to be WN and imposing the HT (or first-order ACF) of the two-sided HP(14400) filter.
  • Figure 4: Coefficients of symmetric causal SSA and HP smoothers of length 201, arbitrarily scaled to unit length: the first SSA design (blue line) replicates the holding time of HP, the second SSA design (violet line) replicates the tracking-ability or target correlation of HP.
  • Figure 5: M-SSA smoother for $\delta=0$: first target series (left), second target (middle) and third target (right). Filters applied to first series (blue), second series (red) and third series (green).
  • ...and 9 more figures

Theorems & Definitions (7)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Theorem 2