Are rational expectations in economics purely forwards looking, or can they have backward looking components? Are rational expectations (Muth, Lucas, Kyland and Prescott) in economics purely forwards looking, or can they have backward looking components?
Depends on what exactly you mean by backward looking component. When people talk about agents being 'backward looking' they often mean adaptive expectations, if that is what you have in mind then answer would be no.
However, rational expectations really just require agents to have model consistent expectations (see Snowdon, Vane, &amp; Wynarczyk,(1994). A modern guide to macroeconomics). So if by 'backward looking components' you mean that agents utilize past information in some way answer is yes. This is especially true if the underlying model predicts that expected variable has some structural relationship to its past realizations. This being said there are not many situations where this occurs.
There can also be mixed expectations models that assume that portion of agents follows adaptive expectations for example and rest rational expectations, but these would not strictly be called rational expectations models.