This paper uses a flexible least squares (FLS) time-varying linear regression technique to investigate coefficient stability for the Goldfeld U.S. money demand model over the period 1959:Q2 to 1985:Q3. Time-paths traced out by the FLS coefficient estimates exhibit shifts in 1974 and 1983; but these shifts are small relative to a persistent downward trend in the estimated coefficient for the inflation rate, indicating potential model misspecification. The FLS estimates also indicate that the "unit root" nonstationarity problem reported by OLS money demand studies disappears if the coefficient estimates are allowed to exhibit even small amounts of time-variation.