FARS: Factor Augmented Regression Scenarios in R

Technical Summary of "FARS: Factor Augmented Regression Scenarios in R"

Problem Statement
In the context of macroeconomic and financial time series, there is a growing demand for predicting the full probability densities of key variables rather than relying solely on point forecasts. This is crucial for assessing macroeconomic vulnerability, designing resilience policies, and constructing realistic stress scenarios to understand economic system resilience under adverse conditions. Existing approaches often face limitations: standard Principal Component (PC) methods struggle when latent factors load only on specific groups of variables (block structures), and few tools allow for the construction of conditional densities under "stressed" factor scenarios while accounting for uncertainty in factor estimation. Furthermore, while Factor-Augmented Quantile Regressions (FA-QRs) can estimate conditional quantiles, integrating them with multi-level factor structures and generating probabilistic stress scenarios requires a unified, flexible framework.

Methodology
The paper presents the FARS package, which implements a four-stage methodology to model and forecast conditional densities:

1. Factor Extraction via Multi-Level Dynamic Factor Models (MLDFM):
   The framework extracts latent factors using a flexible MLDFM structure that accommodates both non-overlapping and overlapping blocks of variables (e.g., geographical regions or economic sectors).

   • Model Structure: The model includes pervasive (global) factors common to all variables, block-specific factors, and potentially semipervasive (pairwise or higher-order) factors that load on specific subsets of blocks.
   • Estimation: Factors are estimated using the Sequential Least Squares (SLS) estimator. The algorithm initializes factors using Canonical Correlation Analysis (CCA) or Principal Components (PC) and iteratively minimizes the residual sum of squares (RSS) by regressing variables on factors and factors on loadings until convergence.
   • Uncertainty Quantification: The package computes asymptotically valid confidence regions for the estimated factors. It accounts for uncertainty in factor loadings and potential cross-correlations in idiosyncratic components by employing a subsampling approach (subsampling subsets of series) to correct the asymptotic Mean Squared Error (MSE). This allows for the construction of confidence ellipsoids representing extreme but plausible factor realizations.

2. Factor-Augmented Quantile Regression (FA-QR):
   The extracted factors serve as regressors in a FA-QR model to estimate the conditional quantiles of a target variable $y_t$. The model is specified as:
   $$q_{\tau^*}(y_{t+h} | y_t, F_t) = \mu(\tau^*, h) + \phi(\tau^*, h)y_t + \sum_{k=1}^r \beta_k(\tau^*, h)F_{kt}$$
   Parameters are estimated using the Koenker and D'Oray algorithm.

3. Conditional Density Construction:
   To recover a full predictive conditional density from the estimated quantiles, the package fits a Skewed-Student (Skew-t) distribution. The parameters of the Skew-t distribution (location, scale, shape, and tail heaviness) are determined by minimizing the squared differences between the estimated FA-QR quantiles and the theoretical quantiles implied by the Skew-t distribution.

4. Stressed Scenario Generation:
   To generate stressed scenarios, the methodology identifies factor values on the boundary of the confidence ellipsoid (defined by a specific stress level $\alpha$) that minimize or maximize a specific quantile of the target variable's conditional distribution. These "stressed factors" are then fed back into the FA-QR model to derive stressed quantiles, which are subsequently smoothed into a stressed conditional density.

Key Contributions
The paper introduces the FARS R package, which offers several distinct technical contributions:

• Flexible Factor Extraction: It is the first published R package to extract factors from MLDFMs with overlapping block structures using the SLS estimator. It supports hierarchical structures with more than two levels (global, semipervasive, and block-specific).
• Inference on Factors: It provides asymptotically valid confidence regions for SLS-extracted factors, explicitly correcting for uncertainty in factor loadings via subsampling, a feature not available in other factor estimation software (which often rely on bootstrapping or ignore loading uncertainty).
• Integrated Density and Risk Framework: It unifies the estimation of FA-QRs, the recovery of conditional densities via Skew-t matching, and the calculation of risk measures (e.g., Growth at Risk, Inflation at Risk).
• Stressed Scenario Design: It implements a unique functionality to derive conditional densities and risk measures under stressed factor scenarios by optimizing factor values on the extreme contours of their joint distribution.
• Modularity: The package is designed to be modular, allowing users to supply their own pre-estimated factors or densities to integrate with the remaining workflow.

Results and Illustrations
The paper illustrates the package's functionality through two empirical applications:

1. European Inflation Risk: The authors extract factors from headline CPI data for 38 European countries, grouped into three geographical blocks (West, East, North). They estimate the conditional density of German inflation. The results show that the "Inflation in Stress" (IiS)—derived under a 99% stress scenario for the factors—is significantly higher than the standard "Inflation at Risk" (IaR), highlighting the importance of accounting for factor stress.
2. US GDP Growth Risk: Following González-Rivera et al. (2024), the authors construct densities for US GDP growth using a dataset of 519 variables divided into three blocks (international macro, domestic macro, international finance), including a pairwise factor between international blocks. The analysis demonstrates that "Growth in Stress" (GiS) is more negative than standard "Growth at Risk" (GaR), indicating that standard average scenarios may underestimate downside risks.

Significance and Claims
The paper claims that the primary significance of FARS lies in its ability to lower the barrier for applied researchers and policy institutions to implement advanced, density-based forecasting methods that account for complex factor structures and uncertainty. By packaging the entire workflow—from flexible factor extraction with overlapping blocks to stress scenario generation—into an open-source R tool, it addresses the computational and implementation challenges previously associated with these methods. The authors emphasize that the package enables the construction of realistic stress scenarios that reveal vulnerabilities (such as higher inflation or lower growth) that would be neglected if only average factor evolution were considered. The paper does not claim to outperform existing methods in all distributional shapes (noting that Skew-t may be less suitable for multimodal distributions) but positions the package as a comprehensive tool for the specific methodology of FA-QRs within MLDFMs.