We introduce a new class of semiparametric dynamic autoregressive models for the Amihud illiquidity measure, which captures both the long-run trend in the illiquidity series with a nonparametric component and the short-run dynamics with an autoregressive component. We develop a GMM estimator based on conditional moment restrictions and an efficient semiparametric ML estimator based on an i.i.d. assumption. We derive large sample properties for our estimators. We further develop a methodology to detect the occurrence of permanent and transitory breaks in the illiquidity process.