Regression Interview Questions
Regression questions in quant interviews test whether you can derive OLS cleanly, name the assumption that just broke, and predict what happens to coefficients when the design matrix misbehaves.
It is overwhelmingly a quant-research topic — and the format is derive-and-justify, not multiple choice.
Last updated 2 July 2026 · sub-areas, difficulty mix and firm attributions on this page are compiled from the 148 regression problems in the QuantVault bank and the firm/role tags on them — exact counts from our own data, no external statistics; no individual author is named in our source data.
- Core sub-areas
- OLS geometry & fit · assumptions & inference · collinearity & omitted variables · ridge/lasso · quantile & factor models
- Difficulty mix
- 24 easy · 98 medium · 26 hard across the 148 problems in our bank
- Who gets asked
- Quant research above all — 146 of the 148 problems are tagged to the QR track
- Free practice
- 16 problems free to open below, each with a full worked solution
Where regression shows up: quant-research phone screens and onsite stats rounds at systematic funds — in our bank the firms with the most regression-tagged problems are Two Sigma (41) and Citadel (32), with Squarepoint, Jane Street and JPMorgan behind them. Two Sigma interview questions › · Citadel interview questions ›
The surfaceThe regression sub-areas quant interviews test
Almost every regression question is one of five sub-areas. Each row pairs the sub-area with the recurring question shape and a representative type — a flavor drawn from our problem bank, not a leaked firm question — so you can calibrate what “knowing regression” actually means at interview depth.
| Sub-area | Recurring shape | Representative type (flavor, not a real question) |
|---|---|---|
| OLS geometry & goodness of fit | Prove what a fit statistic can and cannot tell you | Show R² is unchanged by rescaling the data, or explain why adjusted R² penalizes an extra regressor. |
| Assumptions, violations & inference | State the assumption → name what breaks → pick the fix | What happens to OLS standard errors when errors are correlated, and how a coefficient confidence interval is actually built. |
| Collinearity & omitted variables | Predict the coefficient when the design misbehaves | Sign the bias when a relevant regressor is dropped, or explain why near-collinear columns blow up coefficient variance. |
| Regularization: ridge & lasso | Compare estimators and derive the shrinkage | Why lasso zeroes coefficients while ridge only shrinks them, argued from the penalty geometry or the SVD. |
| Beyond OLS: quantile & factor models | Extend the loss or the design, then diagnose it | Derive the check-loss estimator for a quantile, or estimate a cross-sectional factor model and defend its diagnostics. |
What's confirmed vs. what varies: the sub-areas and difficulty mix above are exact counts from our own 148-problem bank — they describe the practice surface, not any single firm's syllabus. Which sub-area an interviewer reaches for varies by desk; research-heavy loops go deeper into regularization and factor models than trading tracks ever will.
The patternsSignature regression question patterns
Three shapes generate a large share of regression interviews. Each worked box below is a 60–90 second micro-example in the interview's actual derive-and-justify form, built from a free problem in our bank — the reasoning cadence is the thing being graded.
Pattern 1 — the regression you can't just invert
Takeaway: the slope of y on x is not the reciprocal of the slope of x on y — their product is R², and that one identity resolves the whole family of forward/reverse regression questions.
Shape. “You regress y on x and get slope b. What can the slope of x on y be?” — the flavor behind Range of Reverse Regression Slope Given Forward Slope.
1. Write both slopes in the same currency. Forward: \(b = r\,s_y/s_x\). Reverse: \(b' = r\,s_x/s_y\), where \(r\) is the correlation and \(s_x, s_y\) the standard deviations.
2. Multiply. \(b\,b' = r^2 \le 1\). The naive guess \(b' = 1/b\) holds only when \(|r| = 1\), i.e. a perfect fit.
3. Read off the range. Both slopes share the sign of \(r\), and \(b' = r^2/b\), so for a given \(b > 0\) the reverse slope lies in \((0,\, 1/b]\). The move: express every slope as correlation times a ratio of spreads — attenuation falls out for free.
Pattern 2 — sign the omitted-variable bias before they ask
Takeaway: when a relevant regressor is dropped, the kept coefficient absorbs its effect — bias equals (effect of the omitted variable) × (its regression on the kept one), and the sign logic is the scored step.
Shape. “True model \(y = \beta_1 x_1 + \beta_2 x_2 + \varepsilon\); you fit y on \(x_1\) alone. What do you estimate?” — the flavor behind Omitted Variable Bias and Suppressor Variables.
1. Project the omission. Let \(\delta\) be the slope of \(x_2\) on \(x_1\). Substituting the projection into the true model shows the short regression estimates \(\beta_1 + \beta_2\delta\), not \(\beta_1\).
2. Sign it. Bias \(= \beta_2\delta\): positive when the omitted variable helps y and moves with \(x_1\); zero only if \(\beta_2 = 0\) or the regressors are uncorrelated.
3. Flip the trap. A suppressor is the reverse surprise: adding a variable that barely predicts y can strengthen or flip the other coefficient by cleaning noise out of \(x_1\). The move: bias = β₂δ — state it, sign it, then name the suppressor exception.
Pattern 3 — ridge shrinks, lasso selects: say why, not just that
Takeaway: in the SVD basis, ridge multiplies each component by a smooth factor that squeezes small-signal directions hardest, while lasso soft-thresholds — which is exactly why lasso produces exact zeros and ridge never does.
Shape. “Compare ridge and lasso — and explain the mechanism” — the flavor behind Ridge vs. Lasso Shrinkage in the SVD Basis.
1. Rotate to the SVD. With singular values \(d_i\), ridge rescales the i-th component of the least-squares fit by \(d_i^2/(d_i^2 + \lambda)\) — a smooth shrink, harshest where \(d_i\) is small (the near-collinear directions).
2. Contrast the penalty geometry. Lasso's \(\ell_1\) ball has corners on the axes; in the orthogonal case it soft-thresholds, \(\hat\beta_i = \mathrm{sign}(z_i)\max(|z_i| - \lambda, 0)\), so small coefficients hit exactly zero.
3. Draw the practical line. Ridge wins with many small correlated effects; lasso wins when the truth is sparse and you want selection. The move: one smooth shrink factor vs. one soft threshold — derive both and the “when to use which” answer writes itself.
Free practiceFree regression questions by difficulty
Our bank holds 148 regression problems; the 16 below are free to open, each with a full worked solution — 2 easy, 12 medium and 2 hard, mirroring the bank's medium-heavy mix. Every sub-area in the table above is represented, so a pass through this list touches the whole tested surface.
Easy warmups (2)
- R-Squared Invariance Under TransformationsOLS & fiteasyfree
- Why Use Adjusted R-Squared?OLS & fiteasyfree
Medium — the interview core (12)
- Range of Reverse Regression Slope Given Forward SlopeOLS & fitmediumfree
- OLS Assumptions, Violations, and DiagnosticsAssumptions & inferencemediumfree
- OLS with Correlated ErrorsAssumptions & inferencemediumfree
- OLS Coefficient Confidence IntervalsAssumptions & inferencemediumfree
- Detecting Multicollinearity in OLSCollinearitymediumfree
- Effects of Multicollinearity on RegressionCollinearitymediumfree
- Suppressor Variables in Multiple RegressionCollinearitymediumfree
- Ridge vs. Lasso RegressionRegularizationmediumfree
- Ridge vs. Lasso Regression: Theory and PracticeRegularizationmediumfree
- OLS vs Ridge vs Lasso RegressionRegularizationmediumfree
- Ridge vs. Lasso Shrinkage in the SVD BasisRegularizationmediumfree
- Cross-Sectional Factor Model Estimation and DiagnosticsFactor modelsmediumfree
Hard (2)
- Omitted Variable Bias and Suppressor VariablesCollinearityhardfree
- Quantile Regression: Check Loss, KKT Conditions, and Scalable OptimizationQuantile regressionhardfree
How to read the list: the four regularization items are a deliberate ladder — the ridge-vs-lasso comparison recurs at increasing depth, from the conceptual contrast through the three-way estimator comparison to the SVD-basis proof. The remaining 132 problems in the bank, including most of the hard tail, are in the full interactive playlist.
The planHow to prepare for regression questions
Regression rounds are oral derivations: the interviewer probes “why does this hold” harder than the final formula. Make these six things reflexive, in order.
- Derive OLS two ways. Normal equations from minimizing squared error, and the projection picture — \(\hat y\) is the orthogonal projection of y onto the column space, residuals perpendicular to every regressor. Most identities on this page read off that diagram.
- Own the R² family. What R² measures, what survives rescaling and other transformations, and exactly how adjusted R² charges for an extra regressor.
- Pair each assumption with its failure and diagnostic. Exogeneity, homoskedasticity, independent errors, full rank — for each: the symptom when it breaks, what it does to estimates vs. standard errors, and the standard check.
- Drill the bias arithmetic. Omitted-variable bias with its sign logic, suppressor variables, and why collinearity inflates coefficient variance without biasing the fit.
- Know ridge vs. lasso mechanically. Penalty geometry, the SVD shrink factor vs. soft-thresholding, and a one-line answer for when you'd reach for each.
- Rehearse out loud. Write the derivation, then explain it to a wall in under two minutes — the interview grades the narration, not just the algebra.
FAQRegression interview questions — frequently asked
How important is regression for quant interviews?
Essential for quant research, peripheral for pure trading and software tracks. In our bank, 146 of the 148 regression problems are tagged to the quant-research track. Interviewing for QR at a systematic fund, expect at least one derive-and-justify regression question; on a trading-desk loop, probability and expected value dominate instead.
What regression topics should I focus on?
Four clusters cover most of what gets asked: OLS mechanics and the R² family, the assumption list with violations and diagnostics, collinearity plus omitted-variable bias, and ridge vs. lasso. Quantile regression and factor models form the harder tail, mostly in research-heavy loops — the sub-area table above maps each to its recurring shape.
How hard are regression interview questions?
Mostly medium: 24 easy, 98 medium and 26 hard across our 148-problem bank. A medium here means a few minutes of clean reasoning — signing a bias term or comparing estimators — rather than heavy computation; the hard tail adds proofs and optimization detail like the quantile-regression KKT conditions.
Are these real firm interview questions?
No — we do not publish leaked or verbatim questions. These are representative problems matched to the recurring shapes candidates report, rewritten for clarity, with full worked solutions we author ourselves. For firm-specific context, start from Two Sigma or Citadel.
The bottom line: one projection picture, one bias formula and one shrinkage contrast generate most regression interviews — drill the 16 free problems above until those three moves are reflexive, then scale up in the full playlist.