The increasing popularity of voice-based personal assistants provides new opportunities for conversational recommendation. One particularly interesting area is movie recommendation, which can benefit from an open-ended interaction with the user, through a natural conversation. We explore one promising direction for conversational recommendation: mapping a conversational user, for whom there is limited or no data available, to most similar external reviewers, whose preferences are known, by representing the conversation as a user's interest vector, and adapting collaborative filtering techniques to estimate the current user's preferences for new movies. We call our proposed method ConvExtr (Conversational Collaborative Filtering using External Data), which 1) infers a user's sentiment towards an entity from the conversation context, and 2) transforms the ratings of similar'' external reviewers to predict the current user's preferences. We implement these steps by adapting contextual sentiment prediction techniques, and domain adaptation, respectively. To evaluate our method, we develop and make available a finely annotated dataset of movie recommendation conversations, which we call MovieSent. Our results demonstrate that ConvExtr can improve the accuracy of predicting users' ratings for new movies by exploiting conversation content and external data.