<p>Fixed-effects estimates of exercise on happiness.</p>

The exercise × facility interaction was directionally consistent with the substitution hypothesis but not statistically significant (β = −0.022, p = 0.075). Marginal exercise effects declined from +0.095 in zero-facility communities (p < 0.001) to non-significance where three or more facility types were available. Instrumental variable estimation results (first-stage F = 3231.25; second-stage β = −0.158, p = 0.117).

The IV estimate did not reach statistical significance, providing limited evidence on treatment effect heterogeneity. Four-wave balanced panel estimates confirmed the baseline finding (β = +0.074, p < 0.01). CES-D mediation accounted for only approximately 4.6% of the total effect.</p> <p>Conclusions</p><p>Exercise benefits subjective well-being most where community sports facilities are least available—a directionally consistent substitution pattern.

Facility investment may yield greater marginal returns when targeted at underserved communities.</p></div>