Methods for dealing with Price Expectations in a Model Suppose I have a very simple model of wages:
$w_t = \alpha + \beta_1p_t^e + \beta_2p_{t-1} + \beta_3u_t + \epsilon_t$
What are the ways of dealing with the price expectation?
I've thought about an adaptive expectations model, which I could recursively substitute, i.e.:
$p_t^e = \gamma p_{t-1} + (1 - \gamma)p^e_{t-1}$
Any thoughts of alternatives?
 A: Obviously you could try the "rational expectations" hypothesis, which in this case would argue that
$$p^e_t = E[p_t\mid I_{t-1}]$$
where $I_{t-1}$ is the "information available at $t-1$". I presume you have time series data available for the price. Then, except if your model provides an equation for the determination of the price, you could estimate the conditional expected value from this series and obtain an estimated expression  $\hat E[p_t\mid I_{t-1}]$,
For example, if the price time series is mean-stationary, $p_i = E(p_i) + v_i$, with $v_i$ white noise, and no other information except past levels is to be used, then
$$\hat E[p_t\mid I_{t-1}] = \frac 1{t-1}\sum_{i=1}^{t-1}p_i$$
This creates a series of estimated sample mean from a progressively larger sample. Then you could proceed with estimation of the parameters in your original equation. In such a case you would have
$$ w_t = \alpha + \beta_1\big(\hat E[p_t\mid I_{t-1}]+\eta_{t-1}\big) + \beta_2p_{t-1} + \beta_3u_t + \epsilon_t$$
$$\Rightarrow  w_t = \alpha + \beta_1\hat E[p_t\mid I_{t-1}] + \beta_2p_{t-1} + \beta_3u_t + \varepsilon_t,\;\; \varepsilon_t=\big(\beta_1\eta_{t-1}+\epsilon_t\big)$$
where $\eta_{t-1}$ is the error from the estimation of the expected value. For the specific case above, we have that 
$$E(p_t\mid I_{t-1}) = E(p_t), \;\;\hat E[p_t\mid I_{t-1}] = E(p_t) + \frac 1{t-1}\sum_{i=1}^{t-1}v_i$$
and so 
$$\eta_{t-1} = -\frac 1{t-1}\sum_{i=1}^{t-1}v_i$$
In other words, $\hat E[p_t\mid I_{t-1}]$ is an endogenous regressor, which will deprive you of unbiasedness. But the OLS estimator will still be consistent, because the estimation error from the expected value estimation goes to zero asymptotically.  
Naturally, things get more complicated when the price is not mean-stationary.
