The pedestrian, a 35-year-old man from Knaresborough, was crossing Cambridge Road, near the Cenotaph, at around 6pm on Sunday 27 January 2013, when he was hit by a car.
The vehicle, which was being driven by an elderly man, sustained a broken wing mirror in the incident.
The driver initially stopped at the scene of the collision but left without leaving his details.
Officers are appealing for the driver to come forward to help them with their enquiries. Any witnesses to the collision are also urged to contact the police with information.
Anyone who can help the investigation is urged to contact North Yorkshire Police on 101 – select option 2 – and ask for Traffic Constable Steve James or Harrogate Roads Policing Group.
If you wish to remain anonymous, Crimestoppers can be contacted on 0800 555 111.
Please quote reference 12130016345 when passing information about this incident.