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We explore the long-run demand for M1 based on a dataset comprising 32 countries since 1851. We report six main findings: (1) Evidence of cointegration between velocity and the short rate is widespread. For the United States we detect strong evidence based on three of the adjustments to the standard M1 aggregate originally proposed by Goldfeld and Sichel (1990). (2) Evidence of breaks or time-variation in cointegration relationships is weak to non-existent. (3) For several low-inflation countries the data prefer the specification in the levels of velocity and the short rate originally estimated by Selden (1956) and Latané (1960). This is especially clear for the United States. (4) There is no evidence of non-linearities at low interest rates. (5) If the data are generated by either a Selden-Latané or a semi-log specification, estimation of a log-log specification spuriously causes estimated elasticities to appear smaller at low interest rates. (6) Using the correct money demand specification has important implications for the ability to correctly estimate the welfare costs of inflation.