Is Higher Inflation More Variable? Evidence from 151 Economies, 1996–2025, a Regime-Mixture Proof That the Link Can Arise Without Causation, and an Exact Split into Shock Size and Persistence

Friedman argued in his Nobel lecture that high inflation tends to be variable inflation. Using World Bank annual CPI inflation for 151 economies with at least 28 observations in 1996–2025, I find that the rank correlation between a country's mean inflation and its standard deviation is 0.857, and that the log standard deviation rises one for one with the log mean (slope 0.9954, se 0.0749; 0.7387 among countries with mean inflation of 10% or less). I prove two results that sharpen the interpretation. First, by the law of total variance, a country that spends a share p of years in a high-inflation regime has a mean that rises by pΔ and a between-regime standard deviation of Δ√(p(1−p)), so a cross-country link appears even if inflation in each regime is equally predictable. Second, for a stationary AR(1), ln sd = ln σₑ − ½ ln(1−ρ²), and OLS linearity splits the cross-country slope exactly: 0.8386 comes from larger shocks, 0.0456 from persistence and 0.1113 from the small-sample remainder. Within countries, the absolute AR(1) forecast error next year rises by 0.2115 points per point of current inflation (cluster se 0.0360). India's CPI inflation averaged 6.29% with a standard deviation of 2.78; in 2016–2025 the mean was 4.74% and the standard deviation 1.40, the lowest of its three decades.