# Are Latent Factor Regression and Sparse Regression Adequate?

arXiv: https://arxiv.org/abs/2203.01219 · HTML: https://mlsanity.com/paper/2203.01219/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 82 references that are in the map.
- First posted on arXiv 2 Mar 2022.

## Builds on (most cited first, 16 of 16)

- [Confidence Intervals and Hypothesis Testing for High-Dimensional Regression](https://mlsanity.com/paper/1306.3171/) (2013)
- [Adaptive Huber Regression](https://mlsanity.com/paper/1706.06991/) (2017)
- [The Macroeconomy as a Random Forest](https://mlsanity.com/paper/2006.12724/) (2020)
- [Estimating Number of Factors by Adjusted Eigenvalues Thresholding](https://mlsanity.com/paper/1909.10710/) (2019)
- [Sufficient Forecasting Using Factor Models](https://mlsanity.com/paper/1505.07414/) (2015)
- [High-dimensional Ordinary Least-squares Projection for Screening Variables](https://mlsanity.com/paper/1506.01782/) (2015)
- [Understanding Implicit Regularization in Over-Parameterized Single Index Model](https://mlsanity.com/paper/2007.08322/) (2020)
- [Integrative Factor Regression and Its Inference for Multimodal Data Analysis](https://mlsanity.com/paper/1911.04056/) (2019)
- [High-dimensional simultaneous inference with the bootstrap](https://mlsanity.com/paper/1606.03940/) (2016)
- [Factor-Adjusted Regularized Model Selection](https://mlsanity.com/paper/1612.08490/) (2016)
- [Estimating the error variance in a high-dimensional linear model](https://mlsanity.com/paper/1712.02412/) (2017)
- [Learning Latent Factors from Diversified Projections and its Applications to Over-Estimated and Weak Factors](https://mlsanity.com/paper/1908.01252/) (2019)
- [Macroeconomic Data Transformations Matter](https://mlsanity.com/paper/2008.01714/) (2020)
- [Bridging factor and sparse models](https://mlsanity.com/paper/2102.11341/) (2021)
- [Can Machine Learning Catch the COVID-19 Recession?](https://mlsanity.com/paper/2103.01201/) (2021)
- [Embracing the Blessing of Dimensionality in Factor Models](https://mlsanity.com/paper/1610.07697/) (2016)

## Built on by (most cited first, 5 of 5)

- [Ranking Inferences Based on the Top Choice of Multiway Comparisons](https://mlsanity.com/paper/2211.11957/) (2022)
- [Inference on covariance structure in high-dimensional multi-view data](https://mlsanity.com/paper/2509.02772/) (2025)
- [Robust High-dimensional Tuning Free Multiple Testing](https://mlsanity.com/paper/2211.11959/) (2022)
- [Addressing Dynamic and Sparse Qualitative Data: A Hilbert Space Embedding of Categorical Variables](https://mlsanity.com/paper/2308.11781/) (2023)
- [Extensions of Heterogeneity in Integration and Prediction (HIP) with R Shiny Application](https://mlsanity.com/paper/2310.08426/) (2023)

## Abstract (arXiv)

We propose the Factor Augmented sparse linear Regression Model (FARM) that not only encompasses both the latent factor regression and sparse linear regression as special cases but also bridges dimension reduction and sparse regression together. We provide theoretical guarantees for the estimation of our model under the existence of sub-Gaussian and heavy-tailed noises (with bounded (1+x)-th moment, for all x>0), respectively. In addition, the existing works on supervised learning often assume the latent factor regression or the sparse linear regression is the true underlying model without justifying its adequacy. To fill in such an important gap, we also leverage our model as the alternative model to test the sufficiency of the latent factor regression and the sparse linear regression models. To accomplish these goals, we propose the Factor-Adjusted de-Biased Test (FabTest) and a two-stage ANOVA type test respectively. We also conduct large-scale numerical experiments including both synthetic and FRED macroeconomics data to corroborate the theoretical properties of our methods. Numerical results illustrate the robustness and effectiveness of our model against latent factor regression and sparse linear regression models.