In the money demand literature the Linear Quadratic Adjustment Cost (LQAC) model is often considered to explain the observed slow adjustment in agents' money holdings. In this paper we propose a new method of estimating and testing the LQAC model of money demand. In an empirical application to U.K. money demand, we reject the LQAC model defined from a loss function expressed in terms of real money balances. This is in sharp contrast to the results obtained by Cuthbertson and Taylor (1990) in a previous article in this journal using similar techniques on an equivalent data set. In order to explain the discrepancies we point out some pitfalls in the empirical methodology adopted by Cuthbertson and Taylor. We also consider a specification of the LQAC model in terms of nominal money and, despite statistical rejection, we find that if the LQAC model should have any empirical content such a specification should be preferred to the real money specification. (C) Society for Policy Modeling, 1997.