We discuss the economic model and the econometric properties of the Convolution Autoregressive Process of order 1 (C-AR(1)), with focus on the simplest gaussian case. This is a first order autoregressive process in which the error terms are dependent on the lagged value of the process. We show that this dynamics is naturally generated by a model of speculative prices with extrapolative bias in the expectations. Extrapolative expectations bring about excess volatility and excess persistence in the price dynamics. While excess volatility cannot be identified if one only observes the time series of the risk factors, identification can be achieved if the expectations of the variable can be observed from long term forward markets. We show that the model is well suited to generate the excess variance of long maturity prices documented by the Giglio-Kelly variance ratio tests. We finally point out possible extensions of the model beyond the gaussian case both by changing the specification of the expectations model and setting a non linear data generating process for the fundamental process.