At Tessorin, a random forest on 50 features scores a training R-squared of 35% and a test R-squared of -0.8% on daily returns, while a three-feature ridge model scores 0.6% and 0.4%. Explain the gap, choose a model, and say how you would set the number of trees and the depth.

The forest's 35% is memory, not skill: 500 full-depth trees on 375,000 rows can carve about 75,000 leaves each and reproduce the training noise, and the 35.8 point gap to a test R-squared of -0.8% is the measure of that overfit. The ridge's 0.4% out of sample, with a gap of 0.2 points, is a real if small signal: a correlation of about 0.06 with next-day returns. Ship the ridge. If the forest stays, set depth by walk-forward validation, which peaks at depth 4 here, and add trees until the validation score stops moving, around 200.

A student who memorises last year's exam paper scores full marks on it and fails this year's. A tree grown until its leaves are pure keeps splitting until it has isolated the noise of each training day, and with 375,000 stock-days and leaves of 5 observations it can carve about 75,000 leaves per tree; 500 such trees on 50 features have enough freedom to describe the training set almost exactly, so 35% in training says nothing about tomorrow. Daily returns are mostly noise: a 2% daily standard deviation with a true predictable component worth 0.4% of variance means the forecastable part has a standard deviation of only 0.13% a day. A model claiming 35% claims to predict a 1.18% a day component that does not exist. The out-of-sample R-squaredOne minus the ratio of the model squared errors to the squared errors of a constant forecast, measured on data the model never saw. It can be negative. of -0.8% then says the forest's forecasts add error on new data: predicting zero every day would have beaten it.

Judge a model by its test score and by the gap. The ridge's test R-squared of 0.4% is a correlation of about 0.06 with next-day returns, which across 500 stocks every day is a tradeable signal, and its gap of 0.2 points says the training estimate was honest; the forest's test score is negative and its gap is 35.8 points, so it goes nowhere. Three features with four parameters cannot memorise 375,000 rows; that is the point of the ridge penalty and of the small feature set. Two checks before shipping: that the three features were chosen on the training years only, since picking them with the test year in view is leakage of the same kind the forest commits, and that the 0.4% holds across sub-periods and sectors rather than coming from one quarter. A test R-squared is one number from one year; report it with its range across walk-forward folds.

The two settings do different jobs and are tuned differently. Depth controls how much each tree can memorise, so it is tuned by walk-forward validation inside the training years: Tessorin's sweep peaks at depth 4, where leaves hold thousands of stock-days and a split has to improve the fit on a large sample to be made; beyond that, training R-squared keeps climbing and validation falls to -0.8% at full depth. The number of trees does not overfit: each tree is a noisy forecast and averaging more of them lowers the variance of the average, so validation improves with count and then flattens, here past about 200 trees. Add trees until the out-of-bag score stops moving, then stop for cost. Set a minimum leaf size as well as a depth, and subsample features per split, so no single feature dominates. Then compare the tuned forest with the ridge on the same folds: a depth-4 forest at 0.5% validation is close to the ridge's 0.4%, and the ridge is simpler to explain and to monitor, which decides it. The limitation: the validation scores are themselves noisy at this level, so prefer the simplest model inside one standard error of the best.