What does an OLS regression coefficient mean when it is described as holding the other predictors fixed?

## The partial-effect reading Ordinary least squares fits `y_hat = b0 + b1*x1 + b2*x2 + ... + bk*xk` by choosing the coefficients that minimise the sum of squared differences between observed and fitted outcomes. Each slope `bj` is the estimated average change in the outcome associated with a one-unit increase in `xj` **when the other predictors in the model do not change**. Formally it is the partial derivative of the fitted surface with respect to `xj`: everything else in the equation is a constant while you move `xj` by one. This is why a multiple-regression slope is called a *partial* effect and why a simple one-predictor slope is usually a different number. A simple slope lets everything correlated with the predictor drift along with it; a multiple-regression slope does not. ## Always say the units The units of `bj` are **outcome units per predictor unit**. Price in dollars regressed on floor area in square feet gives dollars per square foot. Test score in points regressed on study time in hours gives points per hour. A coefficient with no units attached is not an interpretation, it is a number. Speaking the units aloud also catches nonsense fast: "each extra square foot is worth 210 dollars" is obviously wrong for a coefficient of 210 if the outcome is in thousands of dollars. ## The bedrooms-and-square-footage example Regress house price on square footage and number of bedrooms. People read the bedrooms coefficient as "what an extra bedroom is worth". It is not. It is the average price gap between two houses **of the same total square footage** that differ by one bedroom. Total space is fixed by construction, so the extra bedroom is carved out of the existing space: the same house with more, smaller rooms. That comparison is genuinely different from the one people imagine (bolting an extension onto the house, which would change square footage too), and it is why such a coefficient is often small, sometimes negative, and always a surprise to a stakeholder who has not been told which comparison is on offer. The general lesson: a partial effect answers a question about **like-for-like comparisons within the model's predictor set**, and you have to state that set out loud for the number to mean anything. ## What the arithmetic actually does The Frisch-Waugh-Lovell theorem makes this concrete. Regress `xj` on all the other predictors and keep the residuals: that is the part of `xj` the other predictors cannot explain. Now regress the outcome on those residuals. The slope you get is exactly `bj` from the full model. So a multiple-regression coefficient is estimated from the variation in that predictor that remains after the other predictors have taken their share. Two consequences follow immediately. First, if the other predictors explain most of `xj`, very little variation is left and the estimate is imprecise, with a wide standard error. Second, the coefficient is a property of *this* model specification: the same data with a different set of controls produces a different residualised predictor and therefore a different number. "The coefficient on tenure" is not a fact about the data; it is a fact about a model. ## Holding fixed is not intervening The fit compares observations that already happen to share values on the other predictors. Nobody set anything. Whether the resulting comparison also answers a causal question - what would happen if you *changed* the predictor - depends on assumptions about the data-generating process that live entirely outside the arithmetic of the fit, and it is a separate topic from reading the number. There is also a support question. "Holding square footage fixed while adding a bedroom" describes a comparison that must actually exist in the data. If no 600-square-foot house in the sample has four bedrooms, the model's prediction for that combination is extrapolation driven by the assumed linear form, not evidence. Before quoting a partial effect, check that the combination of predictor values you are describing is populated. ## Reporting it well A defensible sentence has four parts: the size of the change you are describing, the units, the set of predictors held fixed, and the uncertainty. "Among houses of the same square footage, each additional bedroom is associated with about 4,000 dollars less in price, with a 95% interval from 9,000 less to 1,000 more." That is honest, checkable, and immune to the usual misreadings. One caveat on scope: this single-coefficient reading assumes the predictor enters the model in exactly one term. If a predictor appears in several terms at once, the effect of moving it is spread across those terms and no single coefficient carries it.