Zero-inflated stochastic volatility model for disaggregated inflation data with exact zeros

1. Problem Statement

The paper addresses a critical limitation in modeling disaggregated Consumer Price Index (CPI) data: the prevalence of exact zeros (price staleness) in time-series data.

• Context: While aggregate inflation data rarely exhibits exact zeros, disaggregated data (e.g., specific product categories like electricity or medical services) frequently contains exact zeros due to data collection practices (fixed representative products), regulatory price controls, and infrequent price adjustments by firms.
• The Issue: Standard Stochastic Volatility (SV) and Unobserved Components Stochastic Volatility (UCSV) models assume continuous, normally distributed errors. When applied to data with exact zeros, these models suffer from:
  • Biased Estimates: Trend estimates are biased toward zero.
  • Underestimated Volatility: The models interpret zeros as low-variance Gaussian noise rather than structural zeros, leading to overly conservative and uninformative volatility estimates.
  • Poor Forecasting: Point and interval forecasts are poorly calibrated, particularly when zero-inflation is high.
• Gap: Existing zero-inflated models are often designed for discrete data or purely predictive purposes, lacking the ability to jointly estimate latent trends and time-varying volatilities for continuous, non-stationary real-valued data.

2. Methodology

The authors propose a Zero-Inflated Unobserved Components Stochastic Volatility (Z-UCSV) model, extended to a multivariate setting (Z-MUCSV).

A. Model Structure

The model decomposes the observed time series $y_t$ into two latent processes:

1. The Non-Zero Generating Process (UCSV):
   • A latent continuous variable $y^*_t$ follows a standard UCSV structure: $y^*_t = \theta_t + \epsilon^{(y)}_t$, where $\theta_t$ is the latent trend and $\epsilon^{(y)}_t$ has stochastic volatility $h_t$.
   • The trend $\theta_t$ and log-volatility $h_t$ follow random walks.
2. The Zero-Generating Process (Dynamic Logistic Model):
   • A latent probability $p_t$ determines if the observation is zero.
   • $p_t = \text{logit}^{-1}(\pi_t)$, where $\pi_t$ is a latent state following a random walk.
   • The observation equation is a mixture: $y_t \sim (1-p_t)\delta_{y^*_t} + p_t\delta_0$.
   • This allows the model to distinguish between "true" price changes (non-zero) and "structural" price staleness (zero).

B. Multivariate Extension (Z-MUCSV)

• The model is extended to $K$ components to capture cross-sectional dependencies.
• Order-Invariance: Unlike standard Cholesky decompositions which depend on variable ordering, the authors use an order-invariant parameterization (Chan et al., 2024) where the Cholesky factor matrix $C$ is dense and unrestricted, allowing for flexible co-movement estimation without combinatorial explosion.
• Covariance Structure: The covariance matrix for the non-zero component is modeled as $\Sigma^{(y)}(h_t) = C^{-1} \text{diag}(\exp h_t) (C^{-1})^T$.

C. Posterior Inference (Algorithm)

Due to the intractability of the binomial likelihood with a logistic link, the authors develop a custom Gibbs sampler utilizing Pólya–Gamma augmentation:

1. Pólya–Gamma Augmentation: Introduces auxiliary variables $\omega_t$ to restructure the logistic likelihood into a conditionally Gaussian form. This allows the latent probability states ($\pi_t$) to be sampled jointly using Forward-Filtering Backward-Sampling (FFBS).
2. Data Augmentation: For time points where $y_t=0$, the latent non-zero value $y^*_t$ is imputed from the conditional Gaussian distribution.
3. Sampling Steps: The sampler iterates through:
   • Sampling static parameters (variances) and initial states.
   • Sampling latent trends ($\theta$) and volatilities ($h$) using FFBS.
   • Sampling dynamic probabilities ($\pi$) using the Pólya–Gamma augmented FFBS.
   • Sampling the Cholesky factor $C$ for the multivariate case.

3. Key Contributions

1. Novel Model Formulation: First application of a zero-inflated framework specifically designed for continuous, non-stationary, multivariate time series with stochastic volatility. It generalizes the standard UCSV model by explicitly modeling the zero-generating mechanism.
2. Computational Efficiency: Derivation of an efficient Gibbs sampler using Pólya–Gamma augmentation, enabling exact Bayesian inference for time-varying probabilities and volatilities, avoiding the approximations often used in Dynamic Generalized Linear Models (DGLMs).
3. Order-Invariant Multivariate SV: Implementation of a multivariate SV model that does not rely on arbitrary variable ordering, crucial for high-dimensional CPI data (e.g., 49–85 components).
4. Empirical Validation: Comprehensive application to disaggregated CPI data from four advanced economies (US, UK, Germany, Japan), demonstrating the model's superiority over standard approaches.

4. Results

The model was tested on quarterly disaggregated CPI data (1988–2024) across 49–85 components per country.

• Trend Estimation:
  • Standard UCSV models underestimated trend inflation in sectors with high zero-inflation (e.g., regulated utilities, healthcare), pulling estimates toward zero.
  • Z-UCSV recovered higher, more realistic latent trend inflation (e.g., rising to 5% in German "games of chance" vs. near 0% in standard models) by treating zeros as censored observations rather than low values.
• Volatility Estimation:
  • Standard models produced artificially low volatility estimates during periods of price staleness.
  • Z-UCSV provided smoother, higher, and more accurate volatility estimates by interpolating the "missing" information during zero periods.
• Forecasting Performance:
  • Point Forecasts: Z-MUCSV outperformed standard MUCSV in Mean Absolute Error (MAE), particularly for components with high zero-inflation rates (e.g., Japan, Germany).
  • Interval Forecasts: The zero-inflated model delivered significantly better-calibrated prediction intervals (closer to the 45-degree line in coverage plots), whereas standard models produced intervals that were too narrow or misaligned.
• Aggregation: When aggregating component-wise Z-UCSV estimates, the resulting aggregate trend was less sensitive to short-lived spikes and more reflective of underlying inflationary pressure compared to fitting a UCSV model directly to the aggregate index.

5. Significance and Implications

• Policy Relevance: The model provides central banks and policymakers with a more accurate tool for monitoring inflation. By distinguishing between "no change" due to market friction (zero-inflation) and "no change" due to low inflation, it prevents the underestimation of inflationary risks.
• Measurement Error Correction: It formally accounts for the measurement error inherent in disaggregated data collection (staleness), treating it as a stochastic process rather than a nuisance.
• Robustness: The approach is robust across different economic environments, reverting to standard SV behavior when zero-inflation is low, but activating the zero-inflation mechanism when necessary.
• Future Directions: The authors suggest extending the model to include deterministic volatility feedback (SV-in-mean) and dynamic latent factor structures to better capture co-movements across the vast number of CPI components.

In summary, Han and Irie provide a rigorous Bayesian framework that resolves the bias introduced by exact zeros in inflation data, offering superior estimates of latent trends and volatilities, and improved forecasting accuracy for disaggregated economic time series.