S-LASSO selects stock-return predictors—and the shapes of their relationships

Selecting variables and relationships together

The preprint introduces Specification LASSO, or S-LASSO, to address two connected questions: which variables belong in a statistical model, and how should they enter it? Its financial application examines whether company characteristics predict differences in monthly stock returns through linear relationships, nonlinear patterns or interactions with company size.

A linear relationship assigns a constant slope to a predictor. A nonlinear relationship allows that slope to change. An interaction means that the relationship involving one characteristic depends on another. Treating every predictor the same way can therefore miss useful structure, even when the relevant variables are included.

S-LASSO combines these possibilities in a framework designed for settings with many candidate predictors, potentially more than observations. The goal is an explicit model whose selected relationships can be examined, rather than prediction alone.

How the method works

The procedure starts with candidate linear terms, nonlinear functions and pairwise interactions. The authors approximate nonlinear functions using Hermite basis functions—mathematical building blocks that combine to represent curves. Products of these building blocks represent two-variable interaction surfaces.

The first stage applies ordinary LASSO, a regression technique that penalizes coefficients and can shrink them to zero. The second applies adaptive group LASSO, using the initial estimates to weight penalties on individual linear terms and groups of coefficients representing curves or interactions. Components eliminated entirely in the first stage cannot return in the second.

The selection concerns components, not mutually exclusive labels: a characteristic can contribute both a linear term and a nonlinear component. The authors use the Bayesian Information Criterion to choose tuning parameters in their simulations and empirical analysis.

What the simulations show

The simulation experiments use 200 observations and either 100 or 300 candidate variables. Each simulated outcome contains one linear effect, one nonlinear effect and one interaction. Correlations among predictors are calibrated using the distribution of correlations in the financial dataset. Each setting is repeated 100 times.

In one design, the interaction depends on the sum of two variables rather than their product. With 300 candidates, S-LASSO identifies the true interaction in 87% of replications, compared with 15% for the benchmark based on Freyberger, Neuhierl and Weber, called FNW.

The advantage is not universal. In a second design, deliberately favorable to FNW because the interaction depends on a product, both methods achieve perfect selection of the true effects. FNW has slightly lower mean squared error. These are results from specified simulations, not evidence that S-LASSO will dominate every alternative in practice.

Applying it to historical stock returns

The empirical study uses 102 firm characteristics constructed from CRSP, Compustat and I/B/E/S, alongside monthly stock returns from January 1980 through December 2020. The authors adjust delisting returns and address missing characteristics using conditional mean imputation and weighted least squares within a generalized method of moments framework.

The main application restricts interactions to pairs involving firm size. This is both an economic and computational choice: all possible pairs among 102 characteristics would produce 5,151 candidate interactions before their basis expansions. The appendix says the full rolling-window design and associated Gram matrix become prohibitively large.

Across ten four-year subperiods from 1980–1983 through 2016–2019, cash holdings, one-month momentum and share turnover are selected as linear components every time. One-month momentum, industry momentum and firm size are selected as nonlinear components every time. Thirteen characteristics have a selected size interaction in at least one subperiod.

A supplementary analysis allowing all pairwise interactions within those separate subperiods finds size involved in selected interactions in all ten, and industry momentum in nine. This supports size’s prominence without showing that other interactions are absent.

Forecasting gains and conditional guarantees

For out-of-sample evaluation, the researchers repeatedly estimate models using the preceding four years and predict one-month-ahead returns. S-LASSO’s reported mean squared prediction error is 4.03, versus 4.84 for FNW—a 16.7% reduction.

They also construct historical equal-weighted portfolios buying the highest-predicted-return fifth of stocks and shorting the lowest fifth. For the evaluation beginning in 1984, S-LASSO reports a 22.70% annual return and a Sharpe ratio of 1.69, compared with 20.82% and 1.56 for FNW. Maximum drawdowns remain close: 40.08% and 40.29%. These are historical portfolio results, not demonstrated future returns.

The theoretical guarantees are conditional. They require assumptions about sparsity, signal strength, smoothness and dependence among model components. The authors specifically acknowledge that the strong “irrepresentable condition,” needed for their selection-consistency result, can be restrictive with correlated firm characteristics.

Their appendix diagnostics do not verify that condition in the actual application. In one simulated expanded-design setting with 100 candidate variables and 20 relevant variables, the global condition holds in only 20% of replications, despite a high component-by-component satisfaction rate. The authors explicitly caution that the latter does not replace the global requirement.