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Are rational expectations (Muth, Lucas, Kyland and Prescott) in economics purely forwards looking, or can they have backward looking components?

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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, & 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.

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    $\begingroup$ Hi: I'm not clear on what backward looking components means ( but I would probably say yes because expectations are based on the past ), but to get a decent idea of what RE models do, gets Lucas' book : "Studies In Business Cycle Theory". IMHO, he is the clearest of all the RE authors when it comes to academic type RE papers. The book contains most , if not all of his important papers and was a real eye opener for me. $\endgroup$ – mark leeds Aug 8 at 19:26
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    $\begingroup$ I would add to this answer that if the underlying laws of motion are Markovian (a benchmark assumption) or 2-period Markovian (by extending the state), then Forward-looking models will be trivially backward-looking haha. $\endgroup$ – Flintro Aug 9 at 6:23
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    $\begingroup$ @BeckBatucada yes you shouldn't cite stack exchange. This kind expectations are not just exclusive to markow switching models though. Good source is Introduction to Quantitative Macroeconomics Using Julia from Carani (2019) - you can skip all the julia stuff unless you are interested in the new program and just look in the theory and sources cited therein. You should be able to get the book free through university VPN. $\endgroup$ – 1muflon1 Aug 10 at 14:17
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    $\begingroup$ @Beck Batucada: I'm still not clear on the meaning of backward and forward components but I'll try to find notes on the meaning that I was referring to. It's clearly not what you're looking but it's useful to know about because it's talked about a lot in RE. I'll look on the net and get back to you. As far as the rest of the discussion, I'm staying out of it because I don't follow it. $\endgroup$ – mark leeds Aug 10 at 20:30
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    $\begingroup$ I don't think it's relevant to your question (and I don't claim deep understanding of it either ) but, just so you know that I'm not just making this stuff up :), check out part II at this link titled. "Single Stochastic Difference Equations". karlwhelan.com/MAMacro/part6.pdf $\endgroup$ – mark leeds Aug 10 at 20:48

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