Sig·Software Engineer·Technical Phone Screen·Senior

Senior

Apr 2026

Summary

SIG quant researcher interview, technical phone screen that went pretty deep into econometrics fundamentals. One question but it had a lot of moving parts and I definitely underestimated how much they'd push on the theory side.

Questions Asked (1)

Derive the OLS estimator from scratch by minimizing sum of squared residuals, then explain the conditions under which it's BLUE and what efficiency actually means in that context.

Technical Trade-offsData Modeling

Author's notes

I got through the derivation okay, took the gradient, set it to zero, got to (X'X)^{-1} X'y without too much fumbling.

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Suggested Approach

Start by deriving the OLS estimator using matrix calculus: set up the sum of squared residuals, take the derivative with respect to beta, set it to zero, and solve for beta-hat. Then, clearly state the Gauss-Markov assumptions and explain that under these assumptions OLS is BLUE, meaning it has the smallest variance among linear unbiased estimators. Finally, clarify that efficiency refers to this minimum variance property, and discuss how violations of assumptions affect efficiency.

Pro tip: Connect the math to practical implications: for example, mention that heteroskedasticity or autocorrelation breaks the Gauss-Markov assumptions, making OLS inefficient and standard errors biased, which is why robust or GLS methods are used. This shows you understand both theory and real-world application.

1. Set up the minimization problem

Define the linear model y = Xβ + ε and the sum of squared residuals SSR(β) = (y - Xβ)'(y - Xβ). Explain that OLS minimizes this function.

2. Derive the OLS estimator

Take the derivative of SSR with respect to β, set it to zero, and solve the normal equations X'Xβ = X'y, yielding β-hat = (X'X)^{-1}X'y (assuming X'X is invertible).

3. State the Gauss-Markov assumptions

List the assumptions: linearity in parameters, random sampling, no perfect multicollinearity, zero conditional mean (exogeneity), and homoskedasticity with no serial correlation. Explain that these are the conditions for BLUE.

4. Explain BLUE and efficiency

Define BLUE: Best Linear Unbiased Estimator. 'Best' means minimum variance among all linear unbiased estimators. Efficiency here means that OLS achieves the lowest variance, so it produces the most precise estimates under the assumptions.

5. Discuss violations and implications

Mention that if assumptions like homoskedasticity or zero conditional mean fail, OLS may be biased or inefficient. For example, heteroskedasticity makes OLS inefficient, and omitted variable bias violates exogeneity.

Key Points to Mention

Matrix derivation of OLS: normal equations and closed-form solution.
Gauss-Markov assumptions: linearity, exogeneity, homoskedasticity, no autocorrelation, no multicollinearity.
Definition of BLUE: Best Linear Unbiased Estimator.
Efficiency as minimum variance among unbiased estimators.
Consequences of assumption violations: bias, inefficiency, inconsistent standard errors.
Practical relevance: when to use robust standard errors, GLS, or other estimators.

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