Why Distance Fails: Identification Problems and an Alternative Control for Spatial Position

A growing literature documents that distance variables (e.g., distance to CBD, distance to nearest train station, distance to amenity) are not identified in multivariate regressions because the change in distance to one landmark mechanically predetermines the change in distance to every other. A recently proposed fixed-point correction unpacks the Euclidean distance formula into four nested directional terms (Δx, Δx², Δy, Δy²) anchored at any arbitrary reference point, and shows that this construction (i) stabilises all non-correction coefficients against the choice of anchor, (ii) preserves overall model efficiency, and (iii) absorbs the position information that a distance variable imperfectly proxies. We apply the correction to five published studies spanning urban economics, environmental economics, and industrial organisation: Ahlfeldt, Redding, Sturm and Wolf (2015) on the Berlin Wall; Harrison and Rubinfeld (1978) on Boston air quality; Heblich, Redding and Sturm (2020) on the London Underground; Diao, Li, Sing and Zhan (2023) on Singapore's MRT; and Kalnins and LaFontaine (2013) on franchise outlet survival. Three of the five headlines shift by 8 to 49 percent under the correction. In Harrison and Rubinfeld's data, the canonical clean-air willingness-to-pay estimate of $15,927 per unit NOₓ reduction is halved to $8,049, a result that, given the paper's enduring influence on environmental valuation, has independent substantive significance. In Heblich, Redding and Sturm (2020), where parish fixed effects already absorb time-invariant position, the correction is collinear with the existing controls and is correctly inert. In Kalnins and LaFontaine (2013), where the distance variable measures genuinely bilateral spatial structure between an outlet and its own headquarters, the correction also leaves the headline essentially unchanged. We show that when the correction shifts the headline is predictable from the role the distance variable plays in the original specification: it shifts when distance proxies for a broader spatial concept that the model does not otherwise absorb, and is inert when it captures bilateral spatial information that is genuinely orthogonal to position.