29. Machine Learning in Asset Pricing

29.1.1 / 29.1.2 Key Points & Data

Gu et al. (2018) define "machine learning" in asset pricing by three elements: a diverse collection of high-dimensional statistical models, regularization for model selection with overfit mitigation, and efficient algorithms searching among many potential model specifications. They discuss six model classes (below). Data: CRSP 1957.3–2016.12 monthly individual returns for all NYSE/AMEX/NASDAQ firms (~30,000, ~6,200/month); 94 firm characteristics (61 annual/13 quarterly/20 monthly) + 74 industry dummies (SIC); 8 macro predictors (Welch-Goyal 2008: dividend-price ratio dp, earnings-price ep, book-to-market bm, net equity expansion ntis, T-bill rate tbl, term spread tms, default spread dfy, stock variance svar). Conclusions: large predictor sets are feasible for linear prediction with penalization or dimension reduction; allowing nonlinear interactions substantially improves prediction; the best methods are trees and neural networks (shallow beats deep); ML methods are profitable (+26% annualized out-of-sample Sharpe ratio vs buy-and-hold); dominant predictive signals include variations of momentum, liquidity, and volatility.

29.1.3 Setup & Sample Splitting

Stocks \(i=1,\dots,N_t\), months \(t=1,\dots,T\) (balanced panel \(N_t=N\)). Stock \(i\)'s excess return \(r_{i,t+1}\) from \(t\) to \(t+1\), \(P\)-dim raw predictors \(\mathbf z_{i,t}\), additive prediction error \(\epsilon_{i,t+1}\). General form \(r_{i,t+1}=\mathbb E_t[r_{i,t+1}]+\epsilon_{i,t+1}\), where (29.1):

\(g^\star(\cdot)\) a flexible function of predictors, restricted to: the same functional form for all stocks over time (independent of \(i,t\)); depending only on time-\(t\) information (not history) and stock \(i\). Three-way split: training sample (estimate parameters at a specific set of tuning values), validation sample (forecast with training-estimated parameters, compute forecast errors, iterate tuning to maximize the validation objective), testing sample (forecast with the iterated tuning and training-estimated parameters, evaluate out-of-sample OOS performance). Tuning parameters (hyper-parameters) control model complexity (e.g. elastic-net penalty, number of boosting iterations, number and depth of forest trees).

29.1.4 Methodology 1: Simple Linear (OLS)

Linear form \(g^\star(\mathbf z_{i,t})=\mathbf z_{i,t}'\boldsymbol\theta\) (29.2), \(\boldsymbol\theta\) a \(P\times1\) constant (no nonlinear interactions). Objective the standard least squares (29.3): \(\mathcal L(\boldsymbol\theta)=\frac1{NT}\sum_i\sum_t(r_{i,t+1}-g^\star(\mathbf z_{i,t}))^2\); a more robust weighted least squares (29.4) (\(w_{i,t}\) proportional to stock \(i\)'s equity market value at \(t\)); or the Huber robust objective (29.5), \(H(\cdot)\) being (29.6):

The Huber loss assigns squared loss to small errors and absolute loss to large errors (more robust to extreme outliers), \(\xi\) a tuning parameter.

29.1.5 Methodology 2: Penalized Linear

Same model as (29.2), with a penalized objective \(\mathcal L(\boldsymbol\theta;\cdot)=\mathcal L(\boldsymbol\theta)+\phi(\boldsymbol\theta;\cdot)\). Elastic-net penalty:

\(\rho=0\) → lasso (\(L_1\), sets some parameters exactly to zero); \(\rho=1\) → ridge (\(L_2\), pulls values toward but not exactly zero, hence a shrinkage estimator). \(\lambda\) a tuning parameter chosen by iterating between training and validation (allowing \(\lambda\) to differ by predictor can improve out-of-sample performance). See §23.5.3.

29.1.6 Methodology 3: PCR and PLS

PCR two steps: (1) PCA to find a small number \(K\) of principal components (each a linear combination of \(\mathbf z_{i,t}\), see §23.3); (2) run a standard predictive regression on the principal components. Statistical model (29.7): \(\mathbf R_{NT\times1}=(\mathbf Z_{NT\times P}\boldsymbol\Omega_{P\times K})\boldsymbol\theta_{K\times1}+\boldsymbol\epsilon_{NT\times1}\), \(\boldsymbol\Omega\) the weight matrix of the first \(K\) principal components (from eigen-decomposition), \(\boldsymbol\theta\) by OLS. PLS: iteratively construct composite predictors — first regress the forecast target on each raw predictor \(j\) individually for coefficient \(\varphi_j\), then construct a single aggregate predictor as a linear combination of all \(P\) raw factors (weight proportional to \(\varphi_j\), more weight on higher-predicting factors), orthogonalize the target against this aggregate predictor, take the residual to continue, repeating for enough aggregate factors. Weight vector \(\mathbf w_j=\arg\max_{\mathbf w}\text{Cov}^2(r,\mathbf z'\mathbf w)\) s.t. \(\mathbf w'\mathbf w=1\), \(\text{Cov}(\mathbf z'\mathbf w,\mathbf z'\mathbf w_l)=0\) (\(l\neq j\)).

29.1.7 Methodology 4: Generalized Linear

Let \(g^\star(\mathbf z_{i,t})\) be the true model, \(g(\mathbf z_{i,t};\boldsymbol\theta)\) the specified model, \(g(\mathbf z_{i,t};\hat{\boldsymbol\theta})\) the fitted value. Decompose the prediction error:

The intrinsic error \(\epsilon_{i,t+1}\) is irreducible; the estimation error is reducible only by adding more data (not controlled by the econometrician); the approximation error is reducible by adding model flexibility (but too much flexibility causes overfitting and unstable OOS). The model is generalized linear (additive nonlinear basis functions): \(g(\mathbf z;\boldsymbol\theta,\mathbf p(\cdot))=\sum_{j=1}^P\mathbf p(z_j)'\boldsymbol\theta_j\), \(\mathbf p(\cdot)=(p_1(\cdot),\dots,p_K(\cdot))'\) a vector of nonlinear basis functions (e.g. splines). Objective uses the least squares (29.3), extended to include Huber/lasso etc.; the penalty can be group lasso: \(\phi(\boldsymbol\theta;\lambda,L)=\lambda\sum_{j=1}^P(\sum_{k=1}^K\theta_{j,k}^2)^{1/2}\), selecting \((\theta_{j,1},\dots,\theta_{j,K})\) as a group or setting it to zero. \(\lambda,K\) tuning parameters.

29.1.8 Methodology 5: Boosted Trees and Random Forests

Regression trees: a popular fully non-parametric ML approach allowing multi-way predictor interactions. Sequentially slice the space into small regions where data behave similarly; each step sorts the data leftover from the previous step into bins based on one predictor (observations already entering a terminal node no longer participate in further sorting); the final space is sliced into a union of disjoint partitions (terminal nodes), \(g^\star\) approximated by the sample mean of each partition. Model \(g(\mathbf z_{i,t};\boldsymbol\theta,K,L)=\sum_{k=1}^K\theta_k\mathbf 1\{\mathbf z_{i,t}\in C_k(L)\}\), \(K\) the number of terminal partitions (leaves), \(L\) the maximum number of layers (tree depth), \(C_k(L)\) the \(k\)th terminal partition, \(\theta_k\) its parameter. Each terminal partition's \(L_2\) impurity \(H(\theta_k,C_k)=\frac1{|C_k|}\sum_{i,k_{i,t}\in C_k}(r_{i,t+1}-\theta_k)^2\), so \(\theta_k=\frac1{|C_k|}\sum r_{i,t+1}\) is the partition's sample mean.

Boosted regression trees (GBRT): use gradient boosting to ensemble many shallow trees (depth \(L=1\)), outperforming a single complex deep tree. Start from a shallow tree based on one predictor; the residual from the previous shallow tree is shrunk by a factor \(\nu\in(0,1)\) before feeding the next shallow tree (preventing overfitting the residual), the sum of two shallow trees being the new forecast (the first not shrunk), with the residual computed from the updated ensemble; the \(k\)th shallow tree's forecast component is shrunk by \(\nu^{k-1}\); repeat until the number of shallow trees reaches a preset \(B\). Tuning \((L,\nu,B)\). Random forests: ensemble many trees (bootstrap aggregation, bagging). Draw \(B\) bootstrap samples, run a standard (possibly deep) regression tree on each, and to reduce correlation across trees each tree uses only a randomly drawn subset of predictors (dropout), then average the \(B\) results. Benefit: each bootstrap estimate may be deep and overfitting, but averaging stabilizes.

29.1.9 Methodology 6: Neural Networks

A feed-forward neural network includes: an input layer (raw predictors as neurons), hidden layers (zero or more, nonlinear transformations), an output layer (final prediction), and synapses (connections between layers, transmitting signals). Each neuron applies a nonlinear activation function \(f\). The output of the \(i\)th neuron in layer \(k\) (29.8):

The final output is a linear combination of the previous layer's outcomes (29.9): \(g(\mathbf z;\boldsymbol\theta)=\theta_0^{(L-1)}+\sum_{j=1}^{N_{L-1}}\theta_j^{(L-1)}x_j^{(L-1)}\). The authors use ReLU activation \(f(x)=\max\{0,x\}\), so (29.8)→\(x_i^{(k)}=\text{ReLU}((\boldsymbol\theta_i^{(k)})'\mathbf x^{(k-1)})\), (29.9)→ (29.10): \(g(\mathbf z;\boldsymbol\theta)=(\boldsymbol\theta^{(L-1)})'\mathbf x^{(L-1)}\). Consider NN1–NN5: 1 hidden layer of 32 neurons; 2 layers 32,16; 3 layers 32,16,8; 4 layers 32,16,8,4; 5 layers 32,16,8,4,2 (adjacent layers fully connected). The objective is the penalized \(L_2\) prediction error on the training sample \(\mathcal L(\boldsymbol\theta;\cdot)=\frac1{NT}\sum_i\sum_t(r_{i,t+1}-g(\mathbf z;\boldsymbol\theta))^2+\phi(\boldsymbol\theta;\cdot)\), allowing joint updates of all parameters at each step; due to extremely high flexibility, several algorithm tricks are used (see Gu et al. 2018).

29.1.10 / 29.1.11 Metrics

Out-of-sample prediction performance \(R^2_{\text{OOS}}\) (29.11): \(R^2_{\text{OOS}}=1-\frac{\sum_{(i,t)\in\text{test}}(r_{i,t+1}-\hat r_{i,t+1})^2}{\sum_{(i,t)\in\text{test}}r_{i,t+1}^2}\) (the denominator is not demeaned, unlike the traditional \(R^2\)). Model comparison via the Diebold-Mariano statistic (29.12): \(DM_{12}=\frac{\bar d_{12}}{\hat\sigma_{\bar d_{12}}}\), \(d_{12,t}=\frac1{n_{3,t}}\sum_i[(\hat\epsilon_{i,t}^{(1)})^2-(\hat\epsilon_{i,t}^{(2)})^2]\) the mean squared-error difference of two models. Predictor importance: (1) variable importance \(VI_j\) = the reduction in predictive \(R^2_{\text{OOS}}\) when the \(j\)th predictor's coefficient is set to zero (others unchanged); (2) \(VI_j=SSD_j\) (sum of squared partial derivatives) measuring the sensitivity of the final prediction to the \(j\)th predictor.

Out-of-sample \(R^2_{\text{OOS}}\) (%) of each method: neural networks (2–3 layers) and random forests perform well.

("OLS" all raw predictors; "OLS-3" only 3: size, book-to-market, momentum; "+H" Huber loss.) Diebold-Mariano (Fig 29.4): penalization significantly improves unconstrained OLS; neural networks and trees are the only models outperforming "ENet+H"/"GLM+H". Variable importance: among stock characteristics, short-term reversal, momentum, log market equity, and momentum change are most influential, with the least-noisy being accounting variables (dividend initiation/omission, cash flow volatility, etc.); among macro predictors, the most important is the aggregate book-to-market ratio, the least-noisy is market volatility. ML portfolios have high excess returns (Fig 29.5): sort by each model's predicted next-period returns, long the top decile and short the bottom decile (value-weighted), with "NN4" giving the best cumulative return.

29.1.13 / 29.1.14 Contribution & Discussion

Contribution: a comprehensive comparison of relevant ML models' out-of-sample performance in predicting asset returns using standard ML methodology, assembling the most inclusive set of models to date; measuring predictor importance with different methods, guiding future empirical (choosing efficient predictors) and theoretical (inventing economically explanatory models) work; comparing strategies' cumulative returns, with "NN4" best, pointing a path to further optimizing prediction. Discussion: the main concern is that there is no economics in the paper (the authors claim ML-type methods don't have economic intuitions, but this isn't necessarily true); for computational feasibility they strongly assume a time-consistent functional form (constant coefficients of interest), allowing no regime shifting at all, unreasonable over such a long sample; the macro variables are more like aggregate financial-market variables, and should include unemployment, inflation, GDP growth, etc.; nonlinear interactions are important but it isn't shown which interactions matter more; when presenting a strategy's excess returns it's more relevant to show Sharpe ratios than mere risk premium (investors care more about Sharpe ratio), and since the strategy relies on high turnover, the required turnover must be shown feasible in real data. Future directions: improve the algorithm to give parameters economic interpretation; segment the sample by major historical events, estimate separately and compare to judge regime shifting; use macro variables more broadly defined; do pair-wise analysis of all interactions; present in terms of Sharpe ratio and check turnover feasibility.

29.2.1 / 29.2.2 Key Points & Return-Based Characteristics

Moritz-Zimmermann (2016) use the regression tree, an ML model, for asset pricing, with the main body focusing on historical returns as the sole predictor (extensions also consider size, book-to-market, dividend yield, gross profitability, etc.), comparing the tree-sorting model with traditional sorting models and finding tree-based better. Traditional sorting includes: single-variable selection (investing on the portfolio sorted by the best historical variable predicting returns), standard Fama-MacBeth regression (multivariate regression with important predictors), and Fama-MacBeth with variable interactions. The most important historical return predictor is short-term reversal — short-term momentum is crucial for predicting future returns.

General form \(\mathbb E_t[r_{i,t+1}\mid\Theta_{i,t}]=f_t(\Theta_{i,t})\) (29.13), \(\Theta_{i,t}\) the time-\(t\) information set with stock \(i\)'s past information (including predictor values). Define and focus on the return-based predictor \(R_{i,t_f}(g,l)\): \(i\) the individual stock, \(t_f\) the portfolio-formation time, \(g\) the gap between \(t_f\) and the most recent month included in the return calculation, \(l\) the return-calculation horizon length, \(R_{i,t_f}(g,l)\) stock \(i\)'s decile among all stocks. E.g. \(R_{i,t}(1,11)=10\) means stock \(i\) is in the highest decile of returns calculated from 12 months ago to 1 month ago. Return-based strategies always refer to buying the upper decile and selling the lower decile. Baseline predictors include all one-month return deciles in the 2 years before formation, \(g=0,1,\dots,24\), so \(\Theta_{i,t}=\{R_{i,t}(0,1),R_{i,t}(1,1),\dots,R_{i,t}(24,1)\}\) (can recover returns between any two months in the past 2 years).

Baseline model \(f_t(\Theta_{i,t})=a+\sum_{g=0}^{24}\beta_{g,t}R_{i,t}(g,1)\) → (29.14): \(r_{i,t+1}=a+\sum_{g=0}^{24}\beta_{g,t}R_{i,t}(g,1)+\epsilon_{i,t}\). One improvement allowing pairwise interactions of the 25 return predictors is infeasible (needs \(\frac{25\cdot24}2\) predictors), with no theory to stop at two-way interactions.

29.2.3 / 29.2.4 Traditional vs Tree-Based Sorts

Traditional two-level sorting: first branch by some return characteristic \(R(g^{(1)},1)\) with threshold \(\tau^{(1)}\) (\(\le\tau^{(1)}\) to \(a\), $>$ to \(b\)), then a second-level sort by \(R(g^{(2)},1)\) with thresholds for each branch, giving four terminal nodes \(\{S_1,S_2,S_3,S_4\}\), each node's expected return estimated by the sub-sample mean. Tree-based sorting improves on the traditional: thresholds and sorting variables are optimally chosen, sorting can be deeper than two levels, and averaging predictions over many trees stabilizes (significantly improving OOS). With \(L\) terminal nodes \(\{S_1,\dots,S_L\}\), node firms' predicted return \(\hat\mu_l=\text{mean}(r_{i,t+1}\mid\text{Firm }i\in S_l)\), stock \(i\)'s prediction \(\hat r_{i,t+1}=\sum_{l=1}^L\hat\mu_l\mathbf 1\{\text{Firm }i\in S_l\}\) (minimizing the sum of squared errors, maximizing the prediction \(R^2\)). Model averaging (Breiman et al. 1984): each tree uses only a random subset of sorting variables, averaging over \(B\) trees \(\hat r_{i,t+1}=\frac1B\sum_{b=1}^B f_b(\Theta_{i,t})\) (\(B=200\)). Predictor importance: compute the prediction MSE for all stocks using original deciles; randomly permute the \(j\)th predictor and recompute MSE; per tree compute the fraction increase in MSE; average over all trees for the \(j\)th predictor's importance (SSD also possible).

29.2.5–29.2.9 Data, Results, Contribution & Discussion

Data: CRSP 1963–2012 monthly US returns; firm characteristics from Compustat/IBES; only stocks appearing in all datasets (potentially causing sample selection bias); size/value/momentum factors and the risk-free rate from Kenneth French's data library. Results: the equally weighted "strategy" returns from longing the top decile and shorting the bottom decile of predicted returns are high, with the monthly rebalancing portfolio earning positive annual return in all 45 years (Fig 29.6); factor loadings (Fig 29.7): the strategy has a significant but small loading on the market factor and no significant loading on size/value, with the market (and other factors) not explaining the alpha (intercept) much, CAPM and FF3 explaining similarly (\(R^2\) unchanged), and although heavily loading on momentum UMD the alpha is still large (2.05); predictor importance (Fig 29.8): \(R(0,1)\) (last month's return) is most important across the three columns (median/75/25 percentile) → short-term reversal is most crucial. In the appendix, the tree method also beats Fama-MacBeth (even with two-way interactions). Contribution: the first paper in a new field combining ML with empirical asset pricing, an in-depth analysis of historical-return (especially short-term) prediction, contributing to the momentum literature from an ML perspective. Discussion: the main concern is the lack of justification for using past returns as the relevant factors; even discussing exposure to FF3, it's unclear whether past returns are the "right" factors; sample selection bias; should include aggregate factors; should present in Sharpe ratios and check turnover feasibility. Future: include a broader set of firm-characteristic factors and nonlinear interactions (using more advanced ML like neural networks); include aggregate factors and their interactions with characteristic factors; present in Sharpe ratios and check turnover feasibility.

29.3.1 / 29.3.2 Key Points & Tree-Based Sorting

Bryzgalova et al. (2020) use tree ML to generalize conventional sorting, using random forests to predict returns from classical firm characteristics (not return-based) and forming a long-only strategy on the predictions, finding much higher Sharpe ratios than conventional sorting. Difference from Moritz-Zimmermann (2016): the latter uses only return-based predictors, while this paper uses classical firm characteristics. Asset pricing trees (AP-Trees): tree-based sorting can consider a large number of characteristics and their interactions, but for comparison with conventional sorting the paper focuses only on combinations of three variables at a time. A simple decision tree is constructed by a sequence of consecutive splits, each based on one characteristic variable and a threshold. Basis assets: different characteristic variables and different orders of splits give a set of trees (a forest) and corresponding strategies, used to form basis assets — they potentially span the SDF, capture the underlying information in an economically interpretable way, are a good pool of assets for testing other asset pricing models, and the resulting portfolios are well-diversified and don't load on the extreme deciles of cross-sectional sorts.

Shortcomings of conventional (unconditional) sorting into non-overlapping basis assets: the number of sorting characteristics is restricted to 2 or 3 at a time; simple two-way interactions lead to the curse of dimensionality (e.g. size×value two-dim unconditional sort into 5 groups each → 25 Fama-French portfolios, three-dim → 125); the resulting portfolios are not well-diversified, with some intersection cells even empty. Tree-based sorting (conditional on assets available from the last split): first construct a large set of conditional sorts to get all potential building blocks of the SDF; since characteristics are correlated, different orders of variables give completely different trees and portfolios; \(M\) characteristic predictors used in \(d\) steps with two branches each give \(2^d\) portfolios (nodes), \(M^d\) orders, and \(2^d\times M^d\) portfolios total (overlapping, since many trees share stock components); then a shrinkage-based pruning gives an interpretable, sparse, stable subset of basis assets to construct the SDF. Strength: each split is conditional on assets available from the last split, so branches are balanced and well-diversified.

29.3.3 Pruning for Sparsity

The number of portfolios \(2^d\times M^d\) is huge (even larger including all intermediate-step portfolios), so pruning (cutting some portfolios) is needed before constructing the SDF. Existing ML pruning tools are based on local decision criteria, unsuitable for asset pricing which requires a global optimum. The authors use elastic-net shrinkage for a sparse representation of the SDF factors: denote the estimated mean return vector of all \(N\) individual stocks by \(\hat{\boldsymbol\mu}\), estimated variance-covariance by \(\hat{\boldsymbol\Sigma}\), \(\boldsymbol\omega=(\omega_1,\dots,\omega_N)'\) an \(N\times1\) weight vector, objective (29.15):

\(\|\boldsymbol\omega\|_1=\sum|\omega_i|\), \(\|\boldsymbol\omega\|_2^2=\sum\omega_i^2\), three tuning parameters \(\lambda_1\) (lasso weight), \(\lambda_2\) (ridge weight), \(\mu_0\) (target expected return) set arbitrarily on the training sample. Solving gives the most general weight (29.20) (derivation in the collapsible proof):

The pruning tool is applied to tree-sorted portfolios: in any tree, take the pool of portfolios from all nodes (both terminal and intermediate), let investors solve the mean-variance problem from this pool (tuning parameters maximizing the validation Sharpe ratio), and use the testing sample to compare performance.

29.3.4 / 29.3.5 Empirical Analysis & Results

Data: CRSP 1964.1–2016.12 monthly US returns; 10 firm characteristics; one-month T-bill rate as risk-free. Three-way split: training (first 20 years 1964–1983), validation (middle 10 years 1984–1993), testing (last 23 years 1994–2016). Models (3 of 10 characteristics at a time, so \(C_{10}^3=120\) combinations, called 120 cross-sections): tree-based AP-Trees (depth-4 trees; for robustness, exclude trees sorting on the same variable in all 4 steps, avoiding extreme tail portfolios); standard unconditional sorting — Triple Sort (32) (two characteristics into 4 quantiles each, the third (size) only into two halves 50/50), Triple Sort (64) (all three into 4 quantiles). Both AP-Trees and standard sorting use elastic-net estimation to form the mean-variance efficient portfolio (for a fair comparison). Comparison models: FF3, FF5, XSF (4 factors: market + 3 long-short portfolios of that cross-section), FF11 (market + all 10 characteristic long-short portfolios). Evaluation metrics: (1) Sharpe ratio SR (out-of-sample mean-variance efficient portfolio) — higher is better; (2) the \(t\)-statistic of \(\alpha\) (the intercept/mean pricing error of the OOS time-series regression, dependent variable each model's efficient-portfolio return, independent variables the comparison-model factors) — higher \(\alpha\) is better (less spanned by traditional models); (3) cross-sectional adjusted \(R^2\) (XS-\(R^2_{\text{adj}}\)): \(\text{XS-}R^2_{\text{adj}}=1-\frac{N}{N-K}\frac{\sum_i\alpha_i^2}{\sum_i\mathbb E[R_i]^2}\), \(N\) the number of nonzero-weight basis assets, \(K\) the number of comparison-model factors — smaller is better (basis assets jointly more spanned).

Results: (SR) AP-Trees (40 basis assets) beat Triple Sort 32/64 in almost all 120 cross-sections; after pruning, Triple Sort 32/64 beat XSF in almost all cross-sections (Fig 29.9); on \(\alpha\) w.r.t. FF5, AP-Trees win (Fig 29.10); on XS-\(R^2_{\text{adj}}\), AP-Trees win (Fig 29.11/29.12); tree-based sorting with only 10 basis assets (tuning \(\lambda_1\)) suffices to generate higher Sharpe ratios than Triple Sort 32/64 and XSF in most cross-sections (Fig 29.13), with SR's relation to \(\lambda_1\) and \((\lambda_0,\lambda_2)\) (Fig 29.14/29.15, yellow regions high SR, red dot the validation-selected tuning); AP-Trees (10 basis assets) involve a more complicated data-generating process, map to finer return resolution, don't load on extreme portfolios (Fig 29.16 basis-asset weights), and have low turnover (Fig 29.17: AP-Trees monthly/yearly turnover is low, so the high SR doesn't depend on unrealistically high turnover).

29.3.6 / 29.3.7 Contribution & Discussion

Contribution: using characteristic variables commonly used in traditional reduced-form models, it compares tree-based sorting with standard sorting properly with very convincing results; it is among the early attempts to comprehensively compare ML models with traditional reduced-form models, paving the road for future empirical asset pricing with ML. Discussion: the \(\lambda_0\) expression in Appendix A.1 is wrong and should be correctly defined by (29.17); selecting \(\lambda_1\) so that tree-based sorting gives 40 nonzero-weight factors is arbitrary (40 is neither 32 nor 64, and standard unconditional sorting after pruning doesn't necessarily have equal numbers), and future work could match AP-Trees' basis-asset count exactly with Triple Sort 32/64; the sample-splitting is problematic — fixing first 20 training / middle 10 validation / last 23 testing is arbitrary and doesn't rule out regime shifting between any two phases, underestimating predictive power, and a rolling window (the appendix default model) is preferable; always using size as one characteristic and 50/50 for Triple Sort 32 is arbitrary; it's unclear whether the mean-variance efficient portfolio is the right benchmark (investors may care about skewness/kurtosis); it's not really using a decision tree (no tuning of tree parameters like depth/split criteria), so it should avoid calling it "decision trees" and instead "conditional sorts", with future work using a real decision tree letting tree parameters be learned as tuning variables.

29.4.1 / 29.4.2 Key Points & Setup

Monti et al. (2018) study the \(L_1\)-regularized (lasso) linear model for large-scale streaming data. The streaming algorithm must be scalable, amenable to incremental training, and robust to non-stationarity. They focus on how to adaptively select the regularization (lasso) parameter when new data arrives sequentially with a potentially non-stationary distribution (non-stationarity implies the optimal regularization parameter may be time-varying). The adaptive selection is derived for linear regression and extended to generalized linear models; validated with simulation and real data.

For \((\mathbf X_t,y_t)\) arriving sequentially (\(\mathbf X_t\in\mathbb R^p\), \(y_t\in\mathbb R\)), the agent solves \(\boldsymbol\beta\in\mathbb R^p\) to best model \(y_t\) with \(\mathbf X_t\), i.e. \(y_t=\mathbf X_t'\boldsymbol\beta+e_t\). Let \(\lambda\in\mathbb R_+\) be the \(L_1\) lasso penalty; at \(t\), solve the penalized minimization \(\min_{\boldsymbol\beta,\lambda}\mathcal L(\boldsymbol\beta,\lambda)=\sum_{i=1}^t w_i(y_i-\mathbf X_i'\boldsymbol\beta)^2+\lambda\|\boldsymbol\beta\|_1\), \(w_i>0\) proportional to the chronological proximity of the \(i\)th observation to \(t\) (e.g. sliding window or fixed forgetting factor), parameters tuned on a validation sample.

29.4.3 Adaptive Filtering with Fixed Forgetting Factor

Adaptive filtering is the process by which relevant information is assimilated and outdated data discarded, providing a good way to assimilate new information and discard past information without explicitly modeling the data-stream dynamics. Fixed forgetting factor model: let \(r\in(0,1]\) be the fixed forgetting factor; the time-\(t\) sample mean \(\bar{\mathbf X}_t=(1-\frac1{\omega_t})\bar{\mathbf X}_{t-1}+\frac1{\omega_t}\mathbf X_t\), normalizing constant \(\omega_t=\sum_{i=1}^t r^{t-i}=(1+r+\dots+r^{t-1})=r\cdot\omega_{t-1}+1\), sample variance-covariance \(\mathbf S_t=(1-\frac1{\omega_t})\mathbf S_{t-1}+\frac1{\omega_t}(\mathbf X_t-\bar{\mathbf X}_t)'(\mathbf X_t-\bar{\mathbf X}_t)\). The challenge is optimally assigning \(r\); adaptive filtering lets \(r\) be tuned in a data-driven manner (by performance predicting \(\mathbf X_{t+1}\)). Time-varying \(r_t\): define \(C(\mathbf X_{t+1})\) as the residual error in unseen \(t+1\) data \(C(\mathbf X_{t+1})=\mathbf X_{t+1}-\bar{\mathbf X}_t\); if \(\frac{\partial C(\mathbf X_{t+1})}{\partial r}\) can be efficiently calculated, \(r\) updates by stochastic gradient descent (SGD) (29.21): \(r_{t+1}=r_t-\varepsilon\frac{\partial C(\mathbf X_{t+1})}{\partial r}\big|_{r=r_t}\), \(\varepsilon>0\) a fixed step size. (29.21) has the correct direction: when \(\frac{\partial C}{\partial r}<0\) we want to increase \(r_{t+1}\); SGD iteratively updates until the f.o.c. is satisfied.

29.4.4 The Model

Let the lasso penalty parameter be time-varying \(\lambda_t\). Define the objective as the look-ahead negative log-likelihood (29.22): \(C_{t+1}=C(\mathbf X_{t+1},y_{t+1})=\|y_{t+1}-\mathbf X_{t+1}'\hat{\boldsymbol\beta}_t(\lambda_t)\|_2^2\), \(\hat{\boldsymbol\beta}_t\) a function of \(\lambda_t\). If \(\frac{\partial C(\mathbf X_{t+1},y_{t+1})}{\partial\lambda}\big|_{\lambda=\lambda_t}\) can be efficiently calculated, then similar to (29.21), \(\lambda\) updates by (29.23): \(\lambda_{t+1}=\lambda_t-\varepsilon\frac{\partial C(\mathbf X_{t+1},y_{t+1})}{\partial\lambda}\big|_{\lambda=\lambda_t}\) (needs only an initial \(\lambda_0\) and step size \(\varepsilon\)). Once \(\lambda_{t+1}\) is updated, solve \(\hat{\boldsymbol\beta}_{t+1}(\lambda_{t+1})\) (29.24): \(\min_{\boldsymbol\beta}\mathcal L(\boldsymbol\beta,\lambda_{t+1})=\sum_{i=1}^{t+1}w_i(y_i-\mathbf X_i'\boldsymbol\beta)^2+\lambda_{t+1}\|\boldsymbol\beta\|_1\). The difficulty is efficiently computing \(\frac{\partial C}{\partial\lambda}\big|_{\lambda=\lambda_t}\), decomposed by the chain rule (29.25), with \(\hat{\boldsymbol\beta}_t\) closed-form (29.26) and \(\frac{\partial\hat{\boldsymbol\beta}_t}{\partial\lambda}\) closed-form (29.27) (see the collapsible proof).

Real-Time Adaptive Penalization (RAP) algorithm: requires \(\varepsilon\in\mathbb R_+\), \(r_0\in(0,1]\), \(\lambda_0\ge0\). For \(t=1,2,\dots\): (a) receive new \((\mathbf X_{t+1},y_{t+1})\); (b) compute \(\frac{\partial\hat{\boldsymbol\beta}_t}{\partial\lambda}\) by (29.27); (c) get \(\frac{\partial C}{\partial\lambda}\) by (29.25); (d) update \(\lambda_{t+1}\) by (29.23); (e) solve \(\hat{\boldsymbol\beta}_{t+1}(\lambda_{t+1})\) by (29.24); (f) go back to (a). Performance metrics: stationary dataset — track how closely RAP matches the traditional cross-validation (CV) in generating the regularization parameter \(\lambda\), with distance \(\Delta=\|\boldsymbol\beta(\lambda^{\text{CV}})\|_1-\|\boldsymbol\beta(\lambda^{\text{RAP}})\|_1\) (smaller \(\Delta\) better RAP); non-stationary dataset — negative log-likelihood (e.g. 29.22 for an exponential family). Also extended to generalized linear models. Validated with simulation and brain-region connectivity real data.

29.4.5–29.4.7 Results, Contribution & Discussion

Results: simulation shows RAP can efficiently generate the dynamic penalization parameter \(\lambda_t\) in both stationary and non-stationary scenarios; real data uses RAP to show brain-region activity under certain brain tasks. Contribution: focuses on potentially non-stationary streaming data (novel); dynamically updates the lasso penalty \(\lambda\) via the proposed RAP, filling the gap that standard cross-validation in ML is not applicable to non-stationary parameter updating; simulation shows RAP is efficient in both scenarios, practically useful for potentially non-stationary streaming-data problems. Discussion: the step size \(\varepsilon\) (29.21) is arbitrarily fixed, and it's unclear why it can be easily fixed (in the streaming case), so future work could also update \(\varepsilon\) via adaptive filtering; \(\frac{\partial\hat{\boldsymbol\beta}_t}{\partial\lambda}\) in (29.27) should have a \(\frac12\) coefficient; it's unclear whether (29.22) is the correct metric, so future work could generalize (29.22) to depend not just on one future observation but a few to improve stability; the authors use only the \(L_1\) penalty for good reason, but future work should also consider the \(L_2\) penalty, i.e. extend to elastic-net regression and optimally determine both the lasso and ridge penalty parameter values.