What do cluster-robust standard errors change in an OLS fit with many observations per user?

## What the estimator is doing Ordinary least squares gives you coefficients and, separately, an estimate of how uncertain those coefficients are. The default uncertainty calculation assumes every row's error is an independent draw. A cluster-robust variance estimator keeps the first part untouched and replaces the second: instead of building the variance from `n` independent row contributions, it builds it from `G` independent cluster contributions, where each cluster contributes the sum of its own rows before that sum is squared. Whatever correlation exists inside a cluster is therefore carried along rather than assumed to be zero, and only independence *between* clusters is required. This is why the name is a mouthful and the effect is simple. Point estimates: identical. Fitted values, residuals, R-squared: identical. Standard errors, t-statistics, confidence intervals, p-values: recomputed, usually larger. ## Why it usually widens Take 40,000 page views contributed by 800 users, so 50 views per user on average. Users differ persistently — heavy readers, mobile users, people who arrive from one referrer — and those persistent differences make the errors within a user positively correlated. Write the within-user correlation of the outcome as the intraclass correlation `ICC`. For a mean with equal cluster sizes `m`, the variance is inflated by roughly `1 + (m - 1) * ICC`. With `m = 50` and an `ICC` of 0.2 that factor is `1 + 49 * 0.2 = 10.8`, so the effective sample size is about `40,000 / 10.8`, roughly 3,700 rather than 40,000, and the standard error is about `sqrt(10.8)` — a bit over three times — larger than the naive one. Push the correlation to its extreme, where every view from a user is a perfect copy of that user, and the effective sample size falls to the number of users: 800. That ceiling is the intuition to carry into the interview — clustering can never buy you more independent information than you have clusters. The widening is not guaranteed in every case. If a predictor varies freely within a user and is uncorrelated within the cluster, the clustered and naive standard errors can come out similar. Negative within-cluster correlation, which is rarer, can even shrink them. But the direction of the typical surprise is upward, and a clustered standard error that comes out much *smaller* than the naive one deserves a second look rather than a celebration. ## Choosing the level The rule of thumb is to cluster at the level at which the correlation actually operates, and when levels are nested, at the coarsest level that plausibly matters. If users sit inside cities and there is a city-level shock — a marketing campaign, a holiday, an outage — clustering by user leaves that shock unabsorbed and the standard errors remain too small. Clustering by city absorbs both, because a city cluster contains whole users. The cost is that you now have far fewer clusters, and the estimator's reliability depends on that count. Clustering too finely is the common mistake and it fails silently: clustering by page view when the correlation lives at the user level is barely different from doing nothing. ## What it does not do **No efficiency.** The estimator is still ordinary least squares. A model that captures the structure directly, such as a random intercept per user, can be more efficient. Cluster-robust standard errors only stop you from lying about precision. **No bias repair.** If a user-level variable is omitted and correlates with your predictor, the coefficient is wrong and a wider interval around a wrong number does not help. **No rescue from too few clusters.** The theory is asymptotic in `G`, the number of clusters, not in `n`. With a handful of clusters the estimator is downward-biased and the test over-rejects. **No substitute for the right unit.** If the question is really about users, reporting a per-view effect with clustered errors may answer a question nobody asked. ## How to talk about it A strong answer states the invariance first — coefficients unchanged — then the assumption structure — arbitrary correlation within, independence across — then the practical consequence: your precision is governed by the number of clusters. Mentioning that the choice of clustering level is itself a modelling judgment, and that too fine a level quietly reproduces the original problem, is what separates a candidate who has used the tool from one who has read about it.