Momentum and Reversal in Real House Prices: Variance Ratios for 20 Economies and India, 1970-2026

Are house prices a random walk? I use the BIS real residential property price indices for 19 advanced economies and South Africa, quarterly from 1970 to early 2026, and for India from 2009. I prove that the variance ratio at horizon q equals one plus twice a triangle-weighted sum of the first q minus 1 autocorrelations of growth, exactly in the sample when sums are taken circularly, and that for first-order autoregressive growth it converges to (1 plus rho) over (1 minus rho). Real house price growth is strongly persistent: the mean first-order autocorrelation is 0.483 (range 0.081 to 0.733). Mean variance ratios are 2.26 at one year, 3.43 at two years, 4.41 at five years and 4.13 at ten years, against one for a random walk; the heteroskedasticity-robust Lo-MacKinlay test rejects the random walk in all 20 economies at two and five years and in 18 at ten years, and permutation tests agree. Ten-year variance ratios average 1.47 times the first-order autoregressive limit, so persistence extends beyond one quarter. Past four-quarter growth predicts growth over the next four to twelve quarters with slopes of 0.46 to 0.58 but predicts lower growth over the next ten years (slope minus 0.242, standard error 0.074). Past five-year growth predicts lower growth over the next five years (slope minus 0.214), whereas simulated fourth-order autoregressions without level reversion give plus 0.087 on average, and none of 500 simulated panels produces a slope as low as observed (p below 0.002). House prices show short-run momentum and long-run reversal.