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If I have a variable that depends on its expected value in the future among other things (for example inflation), would it be possible to regress it on future values of the dependent variables (in a sense an AR model but with future lags instead). For example runnng the regression $y_t=\beta_0+\beta_1 y_{t+1}+\beta_2 x_{t}+\beta_3 x_{t-1}+\epsilon_t$

Does this create an endogeneity problem? In a sense I am using future values as a proxy for expectations (which would assume agents have rational expectations).

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    $\begingroup$ I believe there are no physical or economic theories that seriously suggest there is any kind of causal relationship in which future values influence present ones. Thus, why not rewrite your model as $$y_{t+1}=-\beta_0/\beta_1-y_1/\beta_1-(\beta_2/\beta_1)x_t-(\beta_3/\beta_1)x_{t-1}-\epsilon_t/\beta_1=\alpha_0+\alpha_1y_t+\alpha_2x_t+\alpha_3x_{t-1}+\delta_t$$with $\operatorname{Var}(\delta_t)=\operatorname{Var}{\epsilon_t}/\beta_1^2$? $\endgroup$
    – whuber
    May 3, 2019 at 14:11
  • $\begingroup$ @Lorenzo: RE modelling is VERY complex. you're thought process is on the right track in that ther are such RE theories but they use future expectations not future realizations. I would either send to economics.se or google for "solutions of rational expectations models". Doing the latter will take you on a journey that you should strap yourself in tightly for. But, if you're really interested in RE, then it will be worth the difficult trip. The former might get you some answers but research on solutions to RE models spans ~50 years. ( early 70's ) so you're not asking a simple question. $\endgroup$
    – mlofton
    May 3, 2019 at 16:54
  • $\begingroup$ @mlofton do you have any specific source that could be useful in understanding how to test RE models? $\endgroup$
    – Lorenzo
    May 4, 2019 at 11:21
  • $\begingroup$ @Lorenzo: there's too much stuff out there. first you have to estimate it correctly and I'm not clear on how you obtained future values in your model. As whuber said, that doesn't make sense. I'll try to find something but it's like walking on a beach and trying to pick out which grain of sand to send you. there's too much stuff. I have a lot and will look to see what might be reasonable. do you have background in rational expectations and what is your academic background ? just the RE concept itself is pretty subtle. took me months just to truly understand it. $\endgroup$
    – mlofton
    May 4, 2019 at 17:40
  • $\begingroup$ @lorenzo: Why don't you send to economics.se first and see what people on there say. I'm over there also so I can still see your responses. $\endgroup$
    – mlofton
    May 4, 2019 at 17:41

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